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INNOVATIVE ASPECTS OF RESEARCH
ADVANCEMENT IN GRAPH THEORY
A Thesis submitted to Gujarat Technological University
for the Award of
Doctor of Philosophy
in
Science-Maths
by
Adalja Divya Ghanshyambhai
(Enrollment No.: 149997673001)
under supervision of
Dr. Gaurang V. Ghodasara
GUJARAT TECHNOLOGICAL UNIVERSITY
AHMEDABAD
FEBRUARY 2020
INNOVATIVE ASPECTS OF RESEARCH
ADVANCEMENT IN GRAPH THEORY
A Thesis submitted to Gujarat Technological University
for the Award of
Doctor of Philosophy
in
Science-Maths
by
Adalja Divya Ghanshyambhai
(Enrollment No.: 149997673001)
under supervision of
Dr. Gaurang V. Ghodasara
GUJARAT TECHNOLOGICAL UNIVERSITY
AHMEDABAD
FEBRUARY 2020
c©Adalja Divya Ghanshyambhai
i
DECLARATION
I declare that the thesis entitled “Innovative aspects of research advancement
in graph theory” submitted by me for the degree of Doctor of Philosophy is the
record of research work carried out by me during the period from May 2015 to
September 2019 under the supervision of Dr. Gaurang V. Ghodasara and this
has not formed the basis for the award of any degree, diploma, associateship, fel-
lowship, titles in this or any other University or other institution of higher learning.
I further declare that the material obtained from other sources has been duly ac-
knowledged in the thesis. I shall be solely responsible for any plagiarism or other
irregularities, if noticed in the thesis.
Signature of Research Scholar : Date : 20/02/2020
Name of Research Scholar : Adalja Divya Ghanshyambhai
Enrollment No. : 149997673001
Place : Rajkot
ii
CERTIFICATE
I certify that the work incorporated in the thesis entitled “Innovative aspects
of research advancement in graph theory” submitted by Ms. Adalja Divya
Ghanshyambhai was carried out by the candidate under my supervision/guidance.
To the best of my knowledge:
(i) the candidate has not submitted the same research work to any other institution
for any Degree/Diploma, Associateship, Fellowship or other similar titles.
(ii) the thesis submitted is a record of original research work done by the Research
Scholar during the period of study under my supervision, and
(iii) the thesis represents independent research work on the part of the Research
Scholar.
Signature of Supervisor : Date : 20/02/2020
Name of Supervisor : Dr. Gaurang V. Ghodasara
Assistant Professor in Mathematics
H. & H. B. Kotak Institute of Science, Rajkot- 360001.
Place : Rajkot
iii
Course-work Completion Certificate
This is to certify that Ms. Adalja Divya Ghanshyambhai, Enrolment number:
149997673001, is a PhD scholar enrolled for PhD program in the branch Science
- Maths of Gujarat Technological University, Ahmedabad.
(Please tick the relevant option(s))
� She has been exempted from the course-work (successfully completed during
M.Phil. Course).
� She has been exempted from Research Methodology Course only (successfully
completed during M.Phil. Course)
� She has successfully completed the PhD course work for the partial requirement
for the award of PhD Degree. Her performance in the course work is as follows:
Grade Obtained in Research Method-
ology (PH001)
Grade Obtained in Self Study Course
(Core Subject) (PH002)
BC AB
Signature of Supervisor :
Name of Supervisor : Dr. Gaurang V. Ghodasara
iv
Originality Report Certificate
It is certified that PhD thesis entitled “Innovative aspects of research advance-
ment in graph theory” by Adalja Divya Ghanshyambhai has been examined by
us. We undertake the following:
(a) Thesis has significant new work/knowledge as compared to already published
or are under consideration to be published elsewhere. No sentence, equation,
diagram, table, paragraph or section has been copied verbatim from previous
work unless it is placed under quotation marks and duly referenced.
(b) The work presented is original and own work of the author (i.e. there is no
plagiarism). No ideas, processes, results or words of others have been presented
as Author’s own work.
(c) There is no fabrication of data or results which have been compiled/analyzed.
(d) There is no falsification by manipulating research materials, equipment or pro-
cesses, or changing or omitting data or results such that the research is not
accurately represented in the research record.
(e) The thesis has been checked using Turnitin (copy of originality report at-
tached) and found within limits as per GTU Plagiarism Policy and instructions
issued from time to time (i.e. permitted similarity index < 10 %).
Signature of Research Scholar : Date : 20/02/2020
Name of Research Scholar : Adalja Divya Ghanshyambhai
Enrollment No. : 149997673001
Place : Rajkot
Signature of Supervisor : Date : 20/02/2020
Name of Supervisor : Dr. Gaurang V. Ghodasara
Place : Rajkot
v
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PhD THESIS Non-Exclusive License to
GUJARAT TECHNOLOGICAL UNIVERSITY
In consideration of being a PhD Research Scholar at GTU and in the interests of the
facilitation of research at GTU and elsewhere, I, Adalja Divya Ghanshyambhai
having Enrollment Number 149997673001 hereby grant a non-exclusive, royalty
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(d) The Universal Copyright Notice c© shall appear on all copies made under the
authority of this license.
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Archives. Any abstract submitted with the thesis will be considered to form
part of the thesis.
(f) I represent that my thesis is my original work, it does not infringe any rights
of others, including privacy rights, and that I have the right to make the grant
conferred by this non-exclusive license.
(g) If third party copyrighted material was included in my thesis for which, under
the terms of the Copyright Act, written permission from the copyright owners is
required, I have obtained such permission from the copyright owners to do the
acts mentioned in paragraph (a) above for the full term of copyright protection.
(h) I retain copyright ownership and moral rights in my thesis, and may deal with
the copyright in my thesis, in any way consistent with rights granted by me to
vii
my University in this non-exclusive license.
(i) I further promise to inform any person to whom I may hereafter assign or
license my copyright in my thesis of the rights granted by me to my University
in this non-exclusive license.
(j) I am aware of and agree to accept the conditions and regulations of PhD in-
cluding all policy matters related to authorship and plagiarism.
Signature of Research Scholar : Date : 20/02/2020
Name of Research Scholar : Adalja Divya Ghanshyambhai
Enrollment No. : 149997673001
Place : Rajkot
Signature of Supervisor : Date : 20/02/2020
Name of Supervisor : Dr. Gaurang V. Ghodasara
Assistant Professor in Mathematics
H. & H. B. Kotak Institute of Science, Rajkot- 360001.
Place : Rajkot
viii
Thesis Approval Form
The viva-voce of the PhD Thesis submitted by Ms. Divya Ghanshyambhai Adalja
(Enrollment No. 149997673001) entitled “Innovative aspects of research ad-
vancement in graph theory” was conducted on
(day and date) at Gujarat Technological University.
(Please tick any one of the following option.)
� The performance of the candidate was satisfactory. We recommend that he/she
shall be awarded the PhD degree.
� Any further modifications in research work recommended by the panel after 3
months from the date of first viva-voce upon request of the Supervisor or request
of Independent Research Scholar after which viva-voce can be re-conducted by the
same panel again.
(briefly specify the modifications suggested by the panel)
� The performance of the candidate was unsatisfactory. We recommend that he/she
should not be awarded the PhD degree.
(The panel must give justifications for rejecting the research work)
Signature of Supervisor with seal :
Name of Supervisor : Dr. Gaurang V. Ghodasara
(External Examiner 1) Name and Signature :
(External Examiner 2) Name and Signature :
(External Examiner 3) Name and Signature :
ix
ABSTRACT
The theory of graphs mainly evolved with the rise of the computer age. It is one
such field of mathematics with cuts across a wide range of disciplines of human
understanding. It has rigorous applications in diversified fields such as computer
science, social sciences, engineering, physics, chemistry and biology. Graphs have
been proved to be a powerful mathematical tool to explain structures of molecules
and it is also possible to explain flow of control with the help of a graph structure.
Development of computer science boost up the research work in this field. There
are many interesting fields of research in graph theory. Decomposition of graphs,
Domination number of graphs, Chromatic graph theory, Theory of hypergraph, Al-
gebraic graph theory, Labeling of graphs and Enumeration of graphs are several
branches of research work in graph theory in various directions.
The field of graph theory has become a field of multifaceted applications ranging
from neural network to bio-technology and coding theory to mention a few. Graphs
are very much useful to solve many problems which are complex in nature but seem-
ingly understandable. The Konigsberg Bridge problem, Four Color problem, Around
the World Game and Traveling Salesman problem are few examples of this character.
Graph labeling is an assignment of numbers/values to vertices or edges or both.
The labeling of graphs is one of the emerging areas of research due to its varied
x
applications. The problems related to labeling of graphs challenge to our mind for
their eventual solutions. In this thesis, we have mainly focused upon the graph
families which satisfy the conditions of different graph labeling techniques such as
divisor cordial labeling, square divisor cordial labeling, cube divisor cordial label-
ing, vertex odd divisor cordial labeling and sum divisor cordial labeling of graphs.
Throughout the thesis, we have considered simple, finite, undirected and connected
graph G = (V,E) with order p and size q.
The present work contains a bonafied record of the research work carried out on
the concepts of graph labeling and contains report of investigations concern to the
concepts of graph labeling.
xi
Acknowledgement
First I sincerely thank The Almighty for the grace showered on me in complet-
ing this research work.
I would like to express my sincere gratitude to my research supervisor and men-
tor Dr. G. V. Ghodasara, Assistant Professor, H. & H. B. Kotak Institute of
Science, Rajkot for his patience, motivation, continuous support, encouragement,
wide experience and immense knowledge to make my research work. His guidance
helped me during the time of research and writing of this thesis.
My special gratitude goes to the Doctorate Progress Committee (DPC) members:
Dr. N. H. Shah, Lecturer, Government Polytechnic, Jamnagar and Dr. N. A.
dani, Senior Lecturer, Government Polytechnic, Rajkot for their precious presence
at every Doctoral Progress Committee (DPC) and providing valuable suggestions
which ensured that the research becomes more significant.
During course of my journey, I have referred many books and good number of
research papers on related topics. I am thankful to the concerned authors.
I would like to express my deep sense of gratitude to Dr R. B. Jadeja, Dr. R.
L. Jhala, the management and the staff members of Marwadi Engineering college
for their sincere support and inspiration during my research journey.
Thanks to my coresearchers Prof. Mitesh Patel and Prof. Mohit Bosmia
for extraordinary help, fruitful suggestions and moral support wherever required.
How can I forget my beloved and respected teachers of school days? The interest
in mathematics grew due to their efforts and expert teaching. I extend deep sense
of gratitude towards them.
I owe my huge debt of thanks to my parents Ghanshyambhai Adalja and
Hiraben Adalja as well as my husband Pinank Patel for their moral support,
encouragement and motivation. Whatever I have achieved in my life is a result of
the blessings and sacrifice of my parents and my husband.
At last I convey my sincere thanks to all those who have provided their kind
support but I have missed to mention them.
Adalja Divya Ghanshyambhai
xii
Contents
List of Nomenclatures xvi
List of Figures xvii
1 Introduction 1
1.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective and scope of work . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Original contribution by the thesis . . . . . . . . . . . . . . . . . . . 3
1.4 Methodology of research . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Achievements with respect to objectives . . . . . . . . . . . . . . . . 5
1.6 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Review of Literature 8
2.1 Historical information of graph theory . . . . . . . . . . . . . . . . . 8
2.2 Basic terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Basic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Operations on a graph / Operations of graphs . . . . . . . . . 11
2.3 Graph labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Divisor Cordial Labeling With The Use of Some Graph Operations 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Some Known Results on DC Labeling . . . . . . . . . . . . . . . . . . 18
3.3 Some New DC Graphs With the Use of Ringsum Operation . . . . . 20
xiii
CONTENTS
3.4 DC Labeling With the Use of Switching Invariance in Cycle Allied
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 DC Labeling With the Use of Duplication of a Vertex/Edge in Star
Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Conclusion and Scope for Further Research . . . . . . . . . . . . . . . 44
4 Square Divisor Cordial, Cube Divisor Cordial and Vertex Odd Di-
visor Cordial Labeling of Graphs 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Square DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Some Known Results on Square DC Labeling . . . . . . . . . 46
4.3 Cube DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.2 Some Known Results on Cube DC Labeling . . . . . . . . . . 49
4.4 Vertex Odd DC Labeling . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.2 Some Known Results on VODC Labeling . . . . . . . . . . . . 50
4.5 New Results on Square DC, Cube DC and Vertex Odd DC Labeling
of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 VODC Labeling With the Use of Switching of a Vertex in Cycle Allied
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.7 VODC Labeling With the Use of Switching of a Vertex in Wheel and
Shell Allied Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.8 VODC Labeling With the Use of Ringsum of Different Graphs with
Star Graph K1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.9 Conclusion and Scope for Further Research . . . . . . . . . . . . . . . 87
5 Sum Divisor Cordial Labeling of Graphs 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Some Existing Results on Sum DC Labeling . . . . . . . . . . . . . . 90
xiv
CONTENTS
5.3 Some New Cycle Related Sum DC Graphs . . . . . . . . . . . . . . . 91
5.4 SDC Labeling of Snakes Related Graphs . . . . . . . . . . . . . . . . 103
5.5 Conclusion and Scope for Further Research . . . . . . . . . . . . . . . 116
6 Sum Divisor Cordial Labeling With the Use of Some Graph Oper-
ations 118
6.1 SDC Labeling of Graphs With the Use of Ringsum of Different Graphs
with Star Graph K1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 SDC Labeling in the Graphs constructed from Corona Product with
K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 SDC Labeling With the Use of Switching of a Vertex in Cycle Allied
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4 SDC Labeling With the Use of Switching of a Vertex in Wheel and
Shell Allied Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.5 SDC Labeling by Duplicating a Vertex/Edge in Star Graph . . . . . . 183
6.6 SDC Labeling by Duplicating Vertex/Edge in Cycle Graph . . . . . . 188
6.7 SDC Labeling by duplicating Vertex/Edge in Path Graph . . . . . . . 196
6.8 Conclusion and Scope for Further Research . . . . . . . . . . . . . . . 203
7 Summary 205
References 208
Annexure 212
xv
CONTENTS
List of Nomenclatures
Nomenclature Meaning
V (G) Vertex set of a graph G
E(G) Edge set of a graph G
|B| Cardinality of set B
d(v) or dG(v) Degree of a vertex v in a graph G
Pn Path graph with n vertices
Cn Cycle graph with n vertices
Kn Complete graph with n vertices
Km,n Complete bipartite graph with m+ n vertices
K1,n Star graph with n+ 1 vertices
Wn Wheel graph with n+ 1 vertices
Gn Gear graph with 2n+ 1 vertices
Sn Shell graph with n vertices
Hn Helm graph with 2n+ 1 vertices
CHn Closed helm graph with 2n+ 1 vertices
Fn Fan graph with n+ 1 vertices
DFn Fan graph with n+ 2 vertices
Fln Flower graph with 2n+ 1 vertices
C(k)n One point union of k copies of cycle Cn
Cn,3 Cycle with twin chords
Cn(1, 1, n− 5) Cycle with triangle
Bn,n Bistar graph with 2n+ 2 vertices
U(m,n) Umbrella graph with m+ n vertices
xvi
CONTENTS
Nomenclature Meaning
Tn Triangular snake
Qn quadrilateral snake
DTn Double triangular snake
DQn Double quadrilateral snake
A(Tn) Alternate triangular snake
A(Qn) Alternate quadrilateral snake
DA(Tn) Double alternate triangular snake
DA(Qn) Double alternate quadrilateral snake
G⊕H Ringsum of two graphs G and H
G⊙
H Corona of two graphs G and H
Pn⊙
K1 Comb graph with 2n vertices
Cn⊙
K1 Crown graph with 2n vertices
ACn Armed crown graph with 3n vertices
G ∪H Union of two graphs G and H
S ′(G) Splitting graph of a graph G
DS(G) Degree splitting graph of a graph G
D2(G) Shadow graph of a graph G
G2 Square of a graph G
N(v) Neighbourhood of vertex v
dne Least integer not less than real number n (Ceiling of n)
bnc Greatest integer not greater than real number n (Floor of n)
DC Divisor Cordial
CDC Cube Divisor Cordial
VODC Vertex Odd Divisor Cordial
SDC Sum Divisor Cordial
W.L.O.G. Without loss of generality
xvii
List of Figures
3.1 DC labeling in W5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 DC labeling in C5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 DC labeling in the graph constructed from ringsum of C6 with one
chord and K1,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 DC labeling in C7,3 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 DC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . . . 25
3.6 DC labeling in W5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.7 DC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 DC labeling in S7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.9 DC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.10 DC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 32
3.11 DC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . 33
3.12 DC labeling in (G6)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.13 DC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 DC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.15 DC labeling in the graph constructed from duplication of vertex by
edge in K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.16 DC labeling in the graph constructed from duplication of edge v0v8
in K1,8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Square DC labeling in S7 . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 CDC labeling in K2,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 VODC labeling in Fl7 . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Square DC labeling in K1,1,6 . . . . . . . . . . . . . . . . . . . . . . . 53
xviii
LIST OF FIGURES
4.5 Square DC labeling in U(9, 3) . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Square DC labeling in C(5)4 . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 Square DC labeling in < K(1)1,5 , K
(2)1,5 > . . . . . . . . . . . . . . . . . . 57
4.8 Square DC labeling in arbitrary supersubdivision of K1,4 . . . . . . . 58
4.9 Square DC labeling in the graph constructed from duplication of an
edge in K1,8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.10 Square DC labeling in K2,5 � u2(K1) . . . . . . . . . . . . . . . . . . 61
4.11 VODC labeling in (G)v. . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.12 VODC labeling in (C8,3)v . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.13 VODC labeling in (C8(1, 1, 3))v . . . . . . . . . . . . . . . . . . . . . 67
4.14 VODC labeling in (W9)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.15 VODC labeling in (G6)v . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.16 VODC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.17 VODC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.18 VODC labeling in the graph constructed from ringsum of C7 with
one chord and K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.19 VODC labeling in C8,3 ⊕K1,8 . . . . . . . . . . . . . . . . . . . . . . 79
4.20 VODC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . 81
4.21 VODC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . 82
4.22 VODC labeling in W6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . 83
4.23 VODC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . 84
4.24 VODC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . 86
4.25 VODC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . 87
5.1 SDC labeling in K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 SDC labeling in C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 SDC labeling in C6 with one chord . . . . . . . . . . . . . . . . . . . 93
5.4 SDC labeling in C7,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 SDC labeling in C8(1, 1, 3) . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 SDC labeling in W5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7 SDC labeling in H6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.8 SDC labeling in Wb5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
xix
LIST OF FIGURES
5.9 SDC labeling in S7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.10 SDC labeling in Fl4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.11 SDC labeling in DF5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.12 SDC labeling in T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.13 SDC labeling in DT5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.14 SDC labeling in Q5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.15 SDC labeling in DQ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.16 SDC labeling in A(T7) . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.17 SDC labeling in A(Q8) . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.18 SDC labeling in DA(T10) . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.19 SDC labeling in DA(Q9) . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1 SDC labeling in C5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 SDC labeling in the graph constructed from ringsum of C7 with one
chord and K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 SDC labeling in C7,3 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . 122
6.4 SDC labeling in C8(1, 1, 3)⊕K1,8 . . . . . . . . . . . . . . . . . . . . 123
6.5 SDC labeling in W6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . . 124
6.6 SDC labeling in Fl4 ⊕K1,4 . . . . . . . . . . . . . . . . . . . . . . . . 126
6.7 SDC labeling in G6 ⊕K1,6 . . . . . . . . . . . . . . . . . . . . . . . . 127
6.8 SDC labeling in P5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . . 128
6.9 DC labeling in S7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.10 SDC labeling in DF5 ⊕K1,5 . . . . . . . . . . . . . . . . . . . . . . . 131
6.11 SDC labeling in K2,7 ⊕K1,7 . . . . . . . . . . . . . . . . . . . . . . . 132
6.12 SDC labeling in K1,6 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 134
6.13 SDC labeling in K2,5 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 135
6.14 SDC labeling in K3,7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 136
6.15 SDC labeling in W7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.16 SDC labeling in H7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.17 SDC labeling in Fl7 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 143
6.18 SDC labeling in F8 �K1 . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.19 SDC labeling in DF6 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 147
xx
LIST OF FIGURES
6.20 SDC labeling in S(K1,5)�K1,5 . . . . . . . . . . . . . . . . . . . . . 148
6.21 SDC labeling in corona of C6 with one chord and K1 . . . . . . . . . 149
6.22 SDC labeling in C7,3 �K1 . . . . . . . . . . . . . . . . . . . . . . . . 150
6.23 SDC labeling in C8(1, 1, 3)�K1 . . . . . . . . . . . . . . . . . . . . . 151
6.24 SDC labeling in (G)v. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.25 SDC labeling in (C8,3)v. . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.26 SDC labeling in (C7(1, 1, 2))v . . . . . . . . . . . . . . . . . . . . . . 159
6.27 SDC labeling in (W9)v1 . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.28 SDC labeling in (G6)v. . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.29 SDC labeling in (S7)v . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.30 SDC labeling in (H6)v. . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.31 SDC labeling in (CH6)v . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.32 SDC labeling in (Fl4)v . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.33 SDC labeling in (B4,5)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.34 SDC labeling in (B4,5)u0 . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.35 SDC labeling in (B4,5)v0 . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.36 SDC labeling in (P5 �K1)v1 . . . . . . . . . . . . . . . . . . . . . . . 176
6.37 SDC labeling in (P5 �K1)u1 . . . . . . . . . . . . . . . . . . . . . . . 176
6.38 SDC labeling in (P5 �K1)u2 . . . . . . . . . . . . . . . . . . . . . . . 177
6.39 SDC labeling in (C7 �K1)v1 . . . . . . . . . . . . . . . . . . . . . . . 178
6.40 SDC labeling in (C7 �K1)u1 . . . . . . . . . . . . . . . . . . . . . . . 178
6.41 SDC labeling in (AC5)v1 . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.42 SDC labeling in (AC5)w1 . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.43 SDC labeling in (AC5)u1 . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.44 SDC labeling in the graph constructed by duplicating edge in K1,8 . . 184
6.45 SDC labeling in the graph constructed by duplicating apex vertex v0
by edge v′0v′′0 in K1,5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.46 SDC labeling in the graph constructed by duplicating vertex v7 by
edge v′7v′′7 in K1,7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.47 SDC labeling in the graph constructed by duplicating an edge by a
vertex in K1,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
xxi
LIST OF FIGURES
6.48 SDC labeling in the graph constructed by duplicating a vertex in C5 . 189
6.49 SDC labeling in the graph constructed by duplicating an edge in C6 . 191
6.50 SDC labeling in the graph constructed by duplicating vertex by edge
in C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.51 SDC labeling in the graph constructed by duplicating each vertex by
new edge in cycle C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.52 SDC labeling in the graph constructed by duplicating edge by new
vertex in C7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.53 SDC labeling in the graph constructed by duplicating a vertex in P5 . 197
6.54 SDC labeling in the graph constructed by duplicating edge in P5 . . . 199
6.55 SDC labeling in the graph constructed by duplicating vertex v′2 by a
new edge v′2v′′2 in P5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.56 SDC labeling in the graph constructed by duplicating an edge v2v3
by a new vertex v′ in P5 . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.57 SDC labeling in the graph constructed by duplicating each vertex by
edge in P5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
xxii
CHAPTER 1
Introduction
This chapter is of introductory nature which gives glimpse of the work embodied
in the thesis and its aim is to provide fundamental components of research. The
essence of results constructed in the other chapters is summarized in this chapter.
1.1 Definition of the problem
The development in the subject of graph labeling is due to painstacking efforts of
many researchers in this field towards solving “Ringel-Kotzig conjecture”. This con-
jecture is considered to be one of the root cause of development in several labeling
techniques like harmonious labeling, cordial labeling, k-equitable labeling etc.
From obtainable written as well as eletronic type literature sources on completely
different topics of graph labeling, some fascinating facts and unsolved problems are
found.
Divisors of a natural number n are the numbers, when divided by n, leaves remainder
0. For example, prime divisors of 210 are 2, 3, 5, 7.
Combining the concepts of divisor of a number from number theory and cordial
labeling from graph labeling, Varatharajan[44] introduced one of the variant of cor-
dial labeling namely DC labeling. Many research papers have been published in this
topic and hence several DC graphs are found. For any two natural numbers a and
b, instead of considering a | b or b | a, if one consider a2 | b or a3 | b then these
will give rise to square DC/cube DC labeling. Some additional changes within the
condition of DC labeling give rise to sum DC labeling and vertex odd DC labeling.
1
1.2. Objective and scope of work
It is interesting to see whether a certain graph which admit one of these labeling
will admit other or not. This thought may produce certain relation/condition be-
tween these invariants of DC labeling. We have focused upon deriving graph families
satisying/not satisfying these invariants of DC labeling.
If a graph satisfies a particular invariant of DC labeling, then “applying different
graph operations, shall the graph preserve that labeling or not” will also be intrest-
ing to see. Here we have considered various graph operations on a certain graph
family and derived the conclusion whether it admits a particular graph labeling or
not. In some cases, we have concluded that some graph operations are labeling
preserving.
In this thesis we have mainly focussed on DC labeling, square DC labeling, cube
DC labeling, vertex odd DC labeling and sum DC labeling. We have considered the
graph operations such as ringsum of different graphs, switching of a vertex, dupli-
cation of vertex/edge, corona product and arbitrary super subdivision.
1.2 Objective and scope of work
The main objective of present research work is to generate new direction to gain
knowledge in the area of graph labeling. Graph labeling is aimed to cover a diversity
of applications in manifold fields. After studying different graph labeling techniques,
the following objectives may be fulfilled.
z Some new graph labeling techniques using/combining the concepts of number
theory, combinatorics and graph theory can be constructed.
z Different graph labelings for the graphs constructed from graph operations can
be investigated.
z Results related to investigations with the use of other graph labeling techniques
and for various graph families can be carried out.
z Using combination of theoretical knowledge and independent mathematical
thinking, one can explore the graphs which satisfy particular graph labeling
technique.
2
1.3. Original contribution by the thesis
z Some new graph labeling techniques may be invented which will give new direc-
tion to young researchers for the development in the field of research in graph
labeling.
z One can identify and explore the family of graphs which satisfy certain graph
labeling techniques but not the other one with particular reason. It helps to
relate different labeling techniques.
z Intensive study of the results derived in this thesis may help to solve conjectures
and open problems.
There is a good scope to investigate equivalent results for different graph families.
Few such problems are stated below.
z Derive essential and adequate condition (if possible) for a graph to become a
DC graph.
z Derive results on DC labeling for generalized petersen graph P (n, k).
z Derive results on DC labeling for line graph, middle graph and total graph of
different graph families.
z Derive results on DC labeling for snake related graphs.
z Derive results on square and cube DC labeling for different operations of two
graphs such as union, ringsum, corona etc.
z Derive results on vertex odd DC labeling for complete lattice grid and related
graphs.
z Derive new graphs on sum DC labeling for line graph, middle graph and total
graph of different graph families.
1.3 Original contribution by the thesis
In the presented research work, different results on DC labeling and its variants
such as square DC labeling, cube DC labeling, vertex odd DC labeling and sum DC
labeling are derived.
DC labeling of graphs:
3
1.3. Original contribution by the thesis
z We have studied some properties of DC graph and we discussed some new DC
graphs by using ringsum operation.
z We have also discussed DC labeling by using duplication of vertex/edge of
special graphs.
z We have established that the graph constructed from switching invariance in
some graph families admit a DC labeling.
Square DC labeling of graphs:
z We have proved that K1,1,n, Um,n, one point union of t copies of the cycle C4,
< K(1)1,n, K
(2)1,n >, arbitrary super subdivision of K1,n, duplication of an edge in
K1,n, K2,n � u2(K1) are square DC graphs.
Cube DC labeling of graphs:
z We have established the results in cube DC labeling similar to the results
established for square DC labeling.
Vertex odd DC labeling of graphs:
z We have studied some properties of vertex odd DC graph and we discuss vertex
odd DC labeling by using ringsum of graphs.
z We also have derived the results of vertex odd DC labeling similar to the results
derived for square DC labeling.
z We have established that the graph constructed from switch of a vertex in some
specific graph families admit a vertex odd DC labeling.
Sum DC labeling of graphs:
z We have proved that cycle, wheel, shell, fan related graphs are sum DC.
z We also have derived that snakes related graphs are sum DC.
z We have discussed sum DC labeling with the use of ringsum and corona product
of two graphs.
z We have studied sum DC labeling by using duplication of vertex/edge in some
graphs.
4
1.4. Methodology of research
z We have established that some graphs constructed from switching invariance
in certain graph families admit sum DC labeling.
1.4 Methodology of research
Initially I started with reading books and research papers based on graph label-
ing which are published in reputed journals. This helped me to develop cohesive
and conceptual thinking. I have adopted some mathematical concepts related to
graph theory and number theory from available sources of literature in printed and
electronic form. Parallelly I focused on LATEX, which is an effective tool for high-
quality typesetting for publication of research work. Later was the study of research
methodology tools such as expansion of past work, modification of mathematical
results, development of new results and efforts to solve conjecture used for further
research work. By refering e-content on combinatorics, I have got proper direction
to work on different graph labeling techniques which is found to be combination
of combinatorics, number theory and graph theory. The concept of divisor of a
number leads to the labeling techniques such as DC labeling, square DC labeling,
cube DC labeling, vertex odd DC labeling and sum DC labeling. Combinatorial and
induction methods have also been used to construct and verify labeling pattern for
a particular graph lebeling defined for the given graph family.
1.5 Achievements with respect to objectives
Since the registration of Ph.D., nine research papers have been published in referred
international journals and one research paper is presented in national conference.
Study of various existing graph labeling techniques for different graph families is a
part of literature survey. We have investigated different graph families which satisfy
DC labeling and main four invariants of it.
1.6 Thesis organization
The thesis consists of seven chapters which delivers the content including some de-
rived results and open problems in different graph labeling techniques.
5
1.6. Thesis organization
Chapter 1 is of introductory nature which gives glimpse of the work embodied
in the thesis and its aim is to provide fundamental components of research in the
present work.
Chapter 2 is basically intended to provide historical information and broad concept
of graph theory as well as essential terminology required for the pertinent work.
Chapter 3 describes DC labeling for some results constructed from different graph
operations such as ringsum of different graphs with star graph, vertex switching and
duplication of vertex.
Chapter 4 aims to discuss square DC, cube DC and vertex odd DC labeling of
graphs. We originate square DC, cube DC and vertex odd DC labeling for some ba-
sic graphs like K1,1,n, one point union C(t)4 of t copies of cycle C4, umbrella U(m,n)
(m,n > 2) etc. We also derive that the graph constructed from arbitrary super
subdivision of K1,n and graph constructed from duplication of an edge in K1,n are
square DC, cube DC and vertex odd DC graphs. We have studied some properties
of vertex odd DC graph and we discuss vertex odd DC labeling for the graphs con-
structed from graph operations ringsum of graphs. We have established that the
graphs constructed from switching of a vertex in some specific graph families admit
vertex odd DC labeling.
Chapter 5 deals with sum DC labeling of graphs. We have proved that some basic
graphs like cycle, wheel, helm, shell, double fan, flower and web are sum DC graphs.
We have also constructed some new snake related sum DC graphs.
Chapter 6 describes sum DC labeling for the graphs constructed from graph oper-
ations such as ringsum of different graphs with star graph K1,n, corona of different
graphs with graph K1, vertex switching of graphs, duplication of a vertex in star,
cycle and path related graphs.
Chapter 7 contains summary of previous chapters. It also includes details of re-
search publications based on the present research work.
Furthermore, at the end of each chapter some open problems based on the con-
templation of results of the current study are presented. These open problems are
6
1.6. Thesis organization
presented with a view to be helpful to young researchers in the field of graph theory.
The references used throughout this work are listed at the end.
7
CHAPTER 2
Review of Literature
This chapter is basically intended to provide historical information and broad idea
of graph theory as well as essential terminology required for the pertinent work.
2.1 Historical information of graph theory
The history of graph theory may be specifically traced from 1735, when Leonhard
Euler (Swiss Mathematician) solved the Konigsberg bridge problem. Euler repre-
sented the first paper in 1736 entitled Solution of a Problem Relating to the Geometry
of Position, which is supposed to be the birth of graph theory. First book on graphs
and related literature was written by Denes Konig in 1936. Another book entitled
Graph Theory was written by Frank Harary in 1969. It was considered the world
over to be the definitive textbook on the subject.
Cayley used graph theory for the study of particular analytical forms. One of the
most famous problems in graph theory is the four color problem which states that,
Is it true that any map drawn in the plane may have regions colored with four colors,
in such a way that any two regions having common border have different colors ?
This problem was first posed by Francis Guthrie in 1852 and first written record of
this problem is in a letter of De Morgan addressed to Hamilton in the same year.
This well celebrated problem took hundred years for its solution. In 1976, Walfgang
Haken and Kenneth Appel solved this problem by giving very lengthy proof.
8
2.2. Basic terminologies
2.2 Basic terminologies
Let G = (V,E) be a graph consists of two finite sets; a non empty set of vertices
V (G) and set of edges E(G) (may or may not be empty). The members of V (G)
and E(G) are commonly termed as graph elements. The cardinality of the vertex
set of a graph G is called order of G whereas the cardinality of its edge set is called
size of G.
Throughout this thesis, we consider a graph G to be simple, undirected, finite and
connected. If |V (G)| = p and |E(G)| = q then we write it as G = (p, q) graph.
2.2.1 Basic graphs
A graph without loops and multiple edges is called a simple graph. The degree
of a vertex v (deg(v) or d(v)) of a graph G is the number of edges incident to
the vertex, with loops counted twice. It is denoted by deg(v) or d(v). A pendant
vertex is a vertex of degree one. A walk in a graph G is a finite sequence W =
v0e1v1e2v2, . . . , vk−1ekvk whose terms are alternately vertices and edges such that,
for 1 ≤ i ≤ k, the edge ei has ends vi−1 and vi. The length of a walk is the number
of edges in it. Path Pn is special walk in which vertex repeatation is not allowed. A
closed path is called a cycle Cn.
Definition 2.2.1 (Bondy and Murty[16]). Wheel Wn is the graph constructed by
join of the graphs Cn and K1. i.e. Wn = Cn +K1. Here the vertices corresponding
to Cn are called rim vertices and Cn is called rim of Wn, while the vertex corresponds
to K1 is called apex vertex.
Definition 2.2.2 (Ma and Feng[22]). Gear Gn is the graph constructed from wheel
Wn by subdividing every of the rim edge of Wn.
Definition 2.2.3 (Deb and Limaye[35]). Shell Sn is the graph constructed as a cycle
Cn with (n − 3) chords sharing a common end point (apex). The shell Sn is same
as fan Fn−1. That is, Sn = Fn−1 = Pn−1 +K1.
Definition 2.2.4 (Ayel and Favaron[20]). Helm Hn is the graph constructed from
the wheel Wn by attaching a pendant edge at every vertex of the rim of Wn.
9
2.2. Basic terminologies
Definition 2.2.5 (Gross and Yellen[17]). Closed helm CHn is the graph constructed
from the helm Hn by joing every pendant vertex to form outer cycle.
Definition 2.2.6 (Gross and Yellen[17]). The web graph Wbn is the graph con-
structed by joining the pendant vertices of helm Hn to form a cycle and then adding
a pendant edge to every vertex of outer cycle.
Definition 2.2.7 (Andar et al.[31]). Flower Fln is the graph constructed from the
helm Hn by attaching every pendant vertex to the apex vertex of helm Hn.
Definition 2.2.8 (Deb and Limaye[35]). The double fan DFn is obtained by Pn +
2K1.
Definition 2.2.9 (Gallian[18]). An edge joining two non-adjacent vertices of cycle
Cn is called chord of cycle Cn.
Definition 2.2.10 (Gallian[18]). Two edges forming a triangle with an edge of the
cycle Cn(n ≥ 5) are called twin chords of a cycle Cn, it is deoted as Cn,3.
Definition 2.2.11 (Gallian[18]). Cycle Cn with three chords which by themselves
form a triangle are called cycle with triangle, it is denoted as Cn(p, q, r) whose edges
form the edges of cycles Cp+2, Cq+2, Cr+2 without chords, where p, q, r, n ∈ N, n ≥ 6
with p+ q + r + 3 = n.
Definition 2.2.12 (Harary[11]). The graph K1,n is called a star graph in which
d(v0) = n called the apex and d(vi) = 1(1 ≤ i ≤ n).
Definition 2.2.13 (Gallian[18]). Bistar Bm,n is the graph constructed by enlinking
the apex vertices of star K1,m and K1,n by an edge.
Definition 2.2.14 (Gallian[18]). The triangular snake Tn is constructed from the
path Pn by replacing every edge of Pn by triangle C3.
Definition 2.2.15 (Gallian[18]). The double triangular snake DTn includes of two
triangular snakes which have a common path.
Definition 2.2.16 (Gallian[18]). The quadrilateral snake Qn is obtained from the
path Pn by replacing every edge of Pn by cycle C4.
10
2.2. Basic terminologies
Definition 2.2.17 (Gallian[18]). The double quadrilateral snake DQn includes of
two quadrilateral snakes that have a common path.
Definition 2.2.18 (Gallian[18]). An alternate triangular snake A(Tn) is obtained
from a path on vertices v1, v2, . . . , vn by joining vi and vi+1 (alternatively) to a new
vertex ui. i.e Every alternate edge of path is replaced by C3.
Definition 2.2.19 (Gallian[18]). An alternate quadrilateral snake A(Qn) is obtained
from a path on vertices v1, v2, . . . , vn by joining vi, vi+1 (alternatively) to new vertices
ui, wi respectively and then joining ui and wi. i.e Every alternate edge of path is
replaced by C4.
Definition 2.2.20 (Gallian[18]). A double alternate triangular snake DA(Tn) con-
sists of two alternate triangular snakes that have a common path. That is, double
alternate triangular snake is obtained from a path on vertices v1, v2, . . . , vn by joining
vi and vi+1 (alternatively) to new vertices ui and wi.
Definition 2.2.21 (Gallian[18]). A double alternate quadrilateral snake DA(Qn)
consists of two alternate quadrilateral snakes that have a common path. That is, it
is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 (alternatively) to new
vertices ui, u′i and wi, w
′i respectively and adding the edges uiwi and u′iw
′i.
Definition 2.2.22 (Gallian[18]). Comb graph Pn�K1 is the graph constructed from
connecting a pendant edge to every vertex of path Pn.
Definition 2.2.23 (Gallian[18]). Crown graph Cn � K1 is the graph constructed
from connecting a pendant edge to every vertex of cycle Cn.
Definition 2.2.24 (Gross et al.[19]). Armed crown is the graph constructed from
connecting path P2 at every vertex of cycle Cn. It is denoted by ACn, Thus ACn =
Cn � P2.
2.2.2 Operations on a graph / Operations of graphs
Definition 2.2.25. Ringsum of two graphs G1 = (V1, E1) and G2 = (V2, E2), de-
noted as G1 ⊕G2, is the graph G1 ⊕G2 = (V1⋃V2, (E1
⋃E2)− (E1 ∩ E2)).
11
2.2. Basic terminologies
Definition 2.2.26 ([18]). Let G = (V,E) be a graph. Let e = uv be an edge of G
and w be a vertex not in G. The edge e is said to be subdivided when it is replaced
by the edges e′ = uw and e′′ = wv.
Definition 2.2.27. Join of two graphs G1 = (V1, E1) and G2 = (V2, E2), denoted as
G1 +G2, is the graph G1 +G2 = (V1⋃V2, (E1
⋃E2)
⋃{uv | u ∈ V (G1), v ∈ V (G2)}.
Definition 2.2.28. The switching of a vertex v in a graph G means removing all
the edges incident to v and adding edges joining v to every other vertex which is not
adjacent to v in G. The graph constructed from switching of a vertex v in a graph
G is denoted as Gv.
Definition 2.2.29. Two adjacent vertices are called neighbours. The set of all
neighbours of vertex v is called the neighbourhood set of v. It is denoted as N(v) or
N [v] and they are respectively known as open and closed neighbourhood sets.
N(v) = {u ∈ V (G) | u adjacent to v and u 6= v}N [v] = N(v)
⋃{v}
Definition 2.2.30 (Harary[11]). Duplication of a vertex v by a new vertex v′ in a
graph G produces a new graph G′, where v ∈ V (G) and v′ is newly added vertex with
N(v) = N(v′).
Definition 2.2.31 (Harary[11]). Duplication of an edge e = uv by a new edge
e′ = u′v′, in a graph G produces a new graph G′, where u, v ∈ V (G) and e′ = u′v′ is
newly added edge with N(u′) = N(u)⋃{v′} \ {v} and N(v′) = N(v)
⋃{u′} \ {u}.
Definition 2.2.32 (Harary[11]). Duplication of a vertex v by a new edge e′ = v′u′
in a graph G produces a new graph G′, where u, v ∈ V (G) and e′ = u′v′ is newly
added edge with N(v′) = {vk, u′} and N(u′) = {vk, v′}.
Definition 2.2.33 (Harary[11]). Duplication of an edge e = uv by a new vertex v′
in a graph G produces a new graph G′, where u, v ∈ V (G) and v′ is newly added
vertex with N(v′) = {u, v}.
Definition 2.2.34 (Frucht and Harary[46]). Corona G�H of two graphs G and H
is defined as the graph acquired by taking one copy of G (having p1 vertices) and p1
copies of H and joining one copy of H at every vertex of G by an edge.
12
2.3. Graph labeling
2.3 Graph labeling
Alexander Rosa introduced graph labeling. In 1966 he introduced certain valuation
of vertices of the graph which is the origin of most of the graph labeling techniques.
The present work is concerned with graph labeling techniques. Different contempla-
tions of combinatorics and number theory give various graph labeling techniques.
The advancement of research in graph labeling techniques is due to basic labelings
known as graceful and harmonious labeling.
Rosa[6] called an injective function f from V (G) to {0, 1, 2, . . . , q} as β-valuation of
graph G, if every edge xy ∈ G allocated by taking the absolute difference of labels
of end vertices are distinct. Later, Golomb[57] called it graceful labeling. Here q
denotes number of edges in the graph G. Graceful labeling in a graceful graph may
not be unique.
In 1980, Graham and Sloane[42] called an injective function f from V (G) to ({0, 1, 2,. . . , q},+q) as harmonious labeling of a graph G if every edge xy ∈ G allocated by
(f(x) + f(y))(mod q) such that every edge is allocated distinct label.
As a bipoduct of efforts to solve Ringel-Kotzig conjecture which states that “All
trees are graceful”, Cahit[14] illuminated the idea of cordial labeling as a weaker
version of graceful and harmonious labeling. A binary vertex labeling f from V (G)
to {0, 1} of a graph G is known as cordial labeling if every edge xy ∈ G allocated
by taking the absolute difference of labels of end vertices and which satisfies the
conditions |vf (1) − vf (0)| ≤ 1 and |ef (1) − ef (0)| ≤ 1. Here vf (i) and ef (i) are
respectively the number of vertices and number of edges of graph G having label i,
i ∈ {0, 1}.
Different variants of graceful and harmonious labeling were studied by different
authors. There are some open problems and conjectures which are the furthermost
13
2.3. Graph labeling
attraction of this field. Cahit[14] derived some basic cordial graphs such as graphs
Km,n, complete graph Kn iff n ≤ 3, fan graph Fn, wheel graph Wn iff n 6≡ 3( mod 4)
and maximal outer planar graphs. Some labelings such as prime cordial labeling,
product cordial labeling, total product cordial labeling, DC labeling, difference cor-
dial labeling etc. were also introduced as different variants of cordial labeling.
A positive integer p(> 1) is said to be prime if it cannot be written as a prod-
uct of two natural numbers a and b for 1 < a, b < p. Oodles number of qualitative
results, properties and conjectures are available for prime numbers. Two integers a
and b are said to be co-prime if (a, b) = 1.
Sundaram, Ponraj and Somasundaram[32] called a vertex labeling f from V (G)
to {1, 2, . . . , |V (G)|} of a graph G as prime cordial labeling if every edge xy ∈ G,
the function f ∗ from E(G) to {0, 1} is defined as f ∗(xy) = 1 if gcd(f(x), f(y)) = 1
and 0 if gcd(f(x), f(y)) > 1 satisfies the condition |ef (1)) − ef (0)| ≤ 1. The same
author proved that Cn iff n ≥ 6, Pn iff n 6= 3, K1,n (n odd), bistars Bn,n, crowns
Cn �K1 are prime cordial graphs[32].
In 2004, Sundaram, Ponraj and Somasundaram[33] called a binary vertex label-
ing f from V (G) to {0, 1} of a graph G as product cordial labeling if every edge
xy ∈ G, the function f ∗ from E(G) to {0, 1} is defined as f ∗(xy) = f(x)f(y) which
fulfils the conditions |vf (1) − vf (0)| ≤ 1 and |ef (1) − ef (0)| ≤ 1. Here vf (i) is the
number of vertices of graph G having label i and ef (i) is the number of edges of
graph G having label i, i ∈ {0, 1}. In [33], they proved that trees, Cn iff n is odd,
triangular snakes Tn and helms Hn are product cordial graphs.
In 2006, Sundaram[34] called a function f from V (G) to {0, 1} of a graph G as
total product cordial labeling if every edge xy ∈ G, the function f ∗ from E(G) to
{0, 1} is defined as f ∗(xy) = f(x)f(y) and it fulfils the conditions |(vf (1) + ef (1))−(vf (0) + ef (0))| ≤ 1.
14
2.3. Graph labeling
Varatharajan, Navaneethakrishnan and Nagarajan[44] called a function f from V (G)
to {1, 2, . . . , |V (G)|} as DC labeling of a graph G if every edge xy ∈ G identified
the label 1 whenever f(x)|f(y) or f(y)|f(x) and 0 otherwise, which satisfies the
condition |ef (1)−ef (0)| ≤ 1. In the same paper, the authors have derived that path
Pn, cycle Cn, wheel Wn, star K1,n, complete graph Kn for n < 7 and graphs K2,n,
K3,n are DC graphs.
In DC labeling, the edge label for an edge ab is produced by using the condition
“whether a divides b or not”; while in square DC labeling, the edge label for an edge
ab is produced by using the condition “whether a2 divides b or not”. Thus the idea
of square DC labeling differs from DC labeling merely by considering square of one
of the end vertices of the centain edge to produce required edge label.
In 2013, S. Murugesan, D. Jayaraman and J. Shiama[56] called a function f :
V (G)→ {1, 2, . . . , |V (G)|} as square DC labeling of a graph G if every edge xy ∈ Gidentified the label 1 whenever [f(x)]2||f(y) or [f(y)]2||f(x) and 0 otherwise, in such
a way that |ef (1) − ef (0)| ≤ 1. In the same paper, the authors have derived that
path Pn, cycle Cn, wheel Wn, star K1,n and some classes of graph Km,n are square
DC graphs.
In 2015, Kanani and Bosmia[25] called a function f from V (G) to {1, 2, . . . , |V (G)|}as cube DC labeling of a graph G if every edge xy ∈ G is identified the label 1
whenever [f(x)]3||f(y) or [f(y)]3||f(x) and 0 otherwise, it satisfies the condition
|ef (1) − ef (0)| ≤ 1. In the same paper, they have derived that path Pn, cycle Cn,
wheel Wn, star K1,n and some classes of Km,n are square DC graphs. The same
authors have constructed cube DC labeling for complete graph Kn, star graph K1,n,
graphs K2,n and K3,n, bistar Bn,n and restricted square graph of Bn,n.
In 2015, Combining by the thought of DC labeling and odd labeling, Muthaiyan
and Pugalenthi[3] called a function f from V (G) to {1, 3, . . . , 2|V (G)|−1} as vertex
odd DC labeling of a graph G if every edge xy ∈ G is identified the label 1 whenever
15
2.4. Concluding Remarks
f(x)|f(y) or f(y)|f(x) and 0 otherwise, it satisfies the condition |ef (1)− ef (0)| ≤ 1.
In the same paper, results on vertex odd DC labeling for shell Sn, helm Hn, flower
Fln, K2,n, < K(1)1,n, K
(2)1,n > were derived.
In 2016, A. Lourdusamy and F. Patrick[1] called a function f from V (G) to {1, 2, . . . ,|V (G)|} as sum DC labeling of a graph G if every edge xy ∈ G identified the label
1 whenever 2|[f(x) + f(y)] and 0 otherwise, in such a way that |ef (1)− ef (0)| ≤ 1.
In this paper, the same authors have established some basic sum DC graphs such
as path Pn, comb Pn�K1, Star K1,n, graph K2,n, bistar Bn,n, jewel, crown Cn�P1,
flower Fln and gear Gn. The authors have also derived sum DC labeling of the
graphs constructed as a result for some graph operations. Few such sum DC graphs
are K2 +mK1, subdivision of the star, K1,3 ∗K1,n and B2n,n.
2.4 Concluding Remarks
This chapter furnishes elementary definitions, terminology and notations necessary
for the advancement of the content. For any undefined notation and terminology
we refer to Harrary[11], Clark and Holton[21], Gross and Yellen[17], West[9], Bondy
and Murty[16], while for the terms related to number theory, we refer to Burton[10].
The next chapter is aimed to the discussion on results related to DC labeling of
graphs resulted due to some graph operations.
16
CHAPTER 3
Divisor Cordial Labeling With
The Use of Some Graph
Operations
Number Theory is a fascinating subject in mathematics. It has so many interest-
ing concepts. The concepts of primality and divisibility play an important role in
number theory. Divisor cordial labeling is one such labeling of graphs which uses
the concept of divisors of a number. Labeling of a graph is allocation of numbers to
vertices or edges or both graph elements. In divisor cordial labeling, the edge labels
are positive integers which are allocated through the condition of divisibility of the
labels (which are of course positive integers) of end vertices.
3.1 Introduction
For a, b ∈ N, a divides b means there exists k ∈ N such that b = ka. It is denoted as
a | b. If a does not divide b, then it is denoted as a - b. By combining the divisibility
concept in number theory and cordial labeling concept in graph labeling, Varathara-
jan, Navaneethakrishnan and Nagarajan originated the notion called divisor cordial
labeling in 2011.
17
3.2. Some Known Results on DC Labeling
Definition 3.1.1 (Varatharajan et al.[44]). Let G = (V,E) be a simple graph with
order p and size q. Consider a bijection f : V (G) → {1, 2, . . . , |V (G)|} and let the
induced function f ∗ : E(G)→ {0, 1} be defined as
f ∗(uv) =
1; if f(u) | f(v) or f(v) | f(u)
0; otherwise
Then the function f is called a divisor cordial labeling if |ef (0)− ef (1)| ≤ 1.
The divisor cordial labeling is also called DC labeling. A graph which confesses DC
labeling is called DC graph.
Example 3.1.1. DC labeling in wheel graph W5 is demonstrated in the following
Figure 3.1.
1
2
6
53
4
v0
v1
v2
v3 v4
v5
Figure 3.1: DC labeling in W5
Naturally, a graph may have more than one divisor cordial labelings. However if
one such labeling exists, the corresponding graph is divisor cordial.
3.2 Some Known Results on DC Labeling
Divisor cordial labeling was introduced in 2011. Since then, many researchers have
explored this labeling by finding captivating results during last eight years.
In first paper on divisor cordial labeling[44], Varatharajan et al. have established
DC labeling for some basic graphs and derived the following results.
Theorem 3.2.1. The path Pn, the cycle Cn and the wheel Wn are DC.
18
3.2. Some Known Results on DC Labeling
Theorem 3.2.2. Kn is DC for n = 1, 2, 3, 5 and 6.
Theorem 3.2.3. Kn is not DC for n = 4 and n ≥ 7.
Theorem 3.2.4. The complete bipartite graphs K1,n, K2,n and K3,n are DC.
Theorem 3.2.5. The barycentric subdivision of the star K1,n is DC.
Theorem 3.2.6. The bistar Bm,n(m ≤ n) is DC.
The same authors have derived more results in [45] which characterize DC graph.
They have also developed DC labeling in some special classes of trees. These results
are listed below.
Theorem 3.2.7. Every full binary tree is DC.
Theorem 3.2.8. The graph G =< K(1)1,n, K
(2)1,n > is DC graph.
Theorem 3.2.9. The graph G =< K(1)1,n, K
(2)1,n, K
(3)1,n > is DC graph.
Many other researchers have worked on this labeling and derived some benchmark
results. Few such results are stated below.
Theorem 3.2.10 (Vaidya and Shah [49]). Splitting graph S ′(K1,n), S ′(Bn,n) of star
K1,n and bistar Bn,n are DC.
Theorem 3.2.11 (Vaidya and Shah [49]). Degree splitting graph DS(Bn,n) and
shadow graph D2(Bn,n) of bistar Bn,n are DC.
Theorem 3.2.12 (Vaidya and Shah [49]). Restricted square graph B2n,n of bistar
Bn,n is DC.
Theorem 3.2.13 (Vaidya and Shah [49]). The helm graph Hn, the flower graph
Fln and the gear graph Gn are DC.
Theorem 3.2.14 (Vaidya and Shah [49]). The graph constructed from switching
invariance in cycle Cn is DC.
Theorem 3.2.15 (Vaidya and Shah [49]). The graph (Wn)v is DC, where v is rim
vertex.
Theorem 3.2.16 (Vaidya and Shah [49]). The graph (Hn)v is DC where v is apex
vertex.
19
3.3. Some New DC Graphs With the Use of Ringsum Operation
Kanani and Bosmia[26] have constructed DC graphs by considering corona product
of K1 with different graph families. In [27], the same authors have discussed DC
labeling for some results constructed from applying different (graph) operations on
bistar Bm,n.
3.3 Some New DC Graphs With the Use of Ringsum Op-
eration
Ghodasara and Rokad[5] illuminated and derived some fascinating results on cordial
labeling of the graphs by considering ringsum of K1,n with different graph families.
Under the inspiration of this credibility, in the current chapter we demonstrate some
new graphs constructed from the graph operation ringsum for DC labeling.
Remark 3.3.1. Throughout this chapter we consider the ringsum of a graph G with
K1,n by considering any one vertex of G and the apex vertex of K1,n as a common
vertex.
Theorem 3.3.1. Cn ⊕K1,n is DC graph.
Proof. Let V (Cn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Cn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}.
It is to be noted that, |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.
Consider a bijection f : V (Cn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 2.
f(v1) = 1.
f(uj) = 2j − 1; 2 ≤ j ≤ n.
f(vk) = 2k; 2 ≤ k ≤ n.
As per this pattern, allocate the vertices such that, for any edge ujuj+1 ∈ E(Cn ⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.
20
3.3. Some New DC Graphs With the Use of Ringsum Operation
Also
f(u1) | f(vk)
for each k, 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = ef (0) = n.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Cn ⊕K1,n is DC graph.
Example 3.3.1. DC labeling in C5 ⊕K1,5 is demonstrated in the following Figure
3.2.
v3 v5v1 v2 v4
1
3
2
4
5
6
7
8
9
10
u1
u2
u3
u5
u4
Figure 3.2: DC labeling in C5 ⊕K1,5
Theorem 3.3.2. G⊕K1,n is DC graph, where G is the cycle Cn with one chord.
Proof. Let G denote the cycle Cn with one chord.
Let V (G ⊕ K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = {uj | 1 ≤ j ≤ n} and
V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u2un},
where u2un is the chord of Cn and vertices u1, u2, un form a triangle with chord u2un.
It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 1.
21
3.3. Some New DC Graphs With the Use of Ringsum Operation
Consider a bijection f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 2.
f(v1) = 1.
f(uj) = 2j − 1; 2 ≤ j ≤ n.
f(vk) = 2k; 2 ≤ k ≤ n.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(G⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.
Also
f(u1) | f(vk)
for each k, 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = n, ef (0) = n+ 1.
Hence, |ef (0)− ef (1)| ≤ 1.
That is, G⊕K1,n is DC graph, where G is the cycle Cn with one chord.
Example 3.3.2. DC labeling of ringsum in C6 with one chord and K1,6 is demon-
strated in the following Figure 3.3.
v3 v4 v5 v6v1 v2
u3
u4
u5
u6
u1
u2 3
2
5
7
9
11
1 4 6 8 10 12
Figure 3.3: DC labeling in the graph constructed from ringsum of C6 with one chord and K1,6
Theorem 3.3.3. Cn,3⊕K1,n is DC graph, where Cn,3 is the cycle with twin chords.
22
3.3. Some New DC Graphs With the Use of Ringsum Operation
Proof. Let V (Cn,3 ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn,3) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Cn,3 ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤
n}⋃{u2un, u2un−1}, where u2un and u2un−1 are the chords of Cn.
It is to be noted that, |V (Cn,3 ⊕K1,n)| = 2n and |E(Cn,3 ⊕K1,n)| = 2n+ 2.
Consider a bijection f : V (Cn,3 ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(uj) = 2j; 1 ≤ j ≤ 2.
f(uj) = 2j + 1; 3 ≤ j ≤ n− 1.
f(un) = 8.
f(vk) = 2k − 1; 1 ≤ k ≤ 3.
f(v4) = 6.
f(vk) = 2k; 5 ≤ k ≤ n.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Cn,3⊕K1,n),
f(uj) - f(uj+1), 2 ≤ j ≤ n− 1.
Also
f(u1) | f(vk), j = 1, 4 ≤ k ≤ n
and
f(u1) | f(u2), f(u1) | f(un).
By looking into the above prescribed pattern,
ef (0) = n+ 1 = ef (1).
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Cn,3 ⊕K1,n is DC graph, where Cn,3 is the cycle Cn with twin chords.
Example 3.3.3. DC labeling in graph C7,3 ⊕K1,7 is demonstrated in the following
Figure 3.4.
23
3.3. Some New DC Graphs With the Use of Ringsum Operation
u3
u4 u5
u6
u7
u1
u2
v3 v4 v5 v6 v7v1 v2
2
7
9 11
3
4
6
8
10 12 14
13
51
Figure 3.4: DC labeling in C7,3 ⊕K1,7
Theorem 3.3.4. Cn(1, 1, n − 5) ⊕ K1,n is DC graph, where Cn(1, 1, n − 5) is the
cycle with triangle.
Proof. Let G be the cycle with triangle Cn(1, 1, n− 5).
Let V (G ⊕ K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = {uj | 1 ≤ j ≤ n} and
V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u1u3,
u3un−1, un−1u1}, where u1, u3 and un−1 be the vertices of the triangle formed by the
chords u1u3, u3un−1 and u1un−1.
It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 3.
Consider a bijection f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 2.
f(v1) = 3.
f(u2) = 1.
f(uj) = 2i− 1; 3 ≤ j ≤ n.
f(vk) = 2j; 2 ≤ k ≤ n.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(G⊕K1,n),
f(uj) - f(uj+1), 3 ≤ j ≤ n− 1.
24
3.3. Some New DC Graphs With the Use of Ringsum Operation
Also
f(u1) | f(vk), 2 ≤ k ≤ n
and
f(u1) | f(u2), f(u2) | f(u3).
By looking into the above prescribed pattern,
ef (0) = n+ 1, ef (1) = n+ 2.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Cn(1, 1, n− 5)⊕K1,n is DC graph.
Example 3.3.4. DC labeling in graph C8(1, 1, 3) ⊕ K1,8 is demonstrated in the
following Figure 3.5.
u3
u4
u5
u6
u7
u1
u2 u8
2
1
5
7
9
11
13
15
3 4 6 8 10 12 1614v3 v4 v5 v6 v7v1 v2 v8
Figure 3.5: DC labeling in C8(1, 1, 3)⊕K1,8
Theorem 3.3.5. Wn ⊕K1,n is DC graph.
Proof. Let V (Wn ⊕K1,n) = {u0, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u0, uj | 1 ≤j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here u0 is the apex vertex, uj(1 ≤ j ≤ n) are rim vertices of Wn and vj(1 ≤ j ≤ n)
are the pendant vertices, u1 is apex vertex of star graph.
Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u1vj |
1 ≤ j ≤ n}.It is to be noted that, |V (Wn ⊕K1,n)| = 2n+ 1 and |E(Wn ⊕K1,n)| = 3n.
25
3.3. Some New DC Graphs With the Use of Ringsum Operation
Consider a bijection f : V (Wn ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 1} defined as below.
f(u0) = 1.
f(uj) = j + 1; 1 ≤ j ≤ n.
f(vk) = n+ k + 1; 1 ≤ k ≤ n.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Wn⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 2.
Also
f(u0) | f(uj), 1 ≤ j ≤ n.
and
f(u1) | f(vk) whenever k is even, 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n2
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = d3n2e, ef (0) = b3n
2c
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Wn ⊕K1,n is DC graph.
Example 3.3.5. DC labeling in W5 ⊕K1,5 is demonstrated in the following Figure
3.6.
v3 v5v1 v2 v4
u1
u2
u3
u5
u4
2
4
6
8 10
31
5
7 9
u0
11
Figure 3.6: DC labeling in W5 ⊕K1,5
26
3.3. Some New DC Graphs With the Use of Ringsum Operation
Theorem 3.3.6. Fln ⊕K1,n is DC graph.
Proof. Let V (Fln ⊕K1,n) = {u, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are
external vertices of Fln and d(vj) = 1, where 1 ≤ j ≤ n, w1 is apex vertex of star
graph.
Let E(Fln ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uuj | 1 ≤ j ≤ n}⋃{uwj | 1 ≤ j ≤n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.
Consider a bijection f : V (Fln ⊕K1,n)→ {1, 2, 3, . . . , 3n+ 1} defined as below.
f(u) = 1.
f(uj) = 2i+ 1; 1 ≤ j ≤ n.
f(wj) = 2j; 1 ≤ j ≤ n.
Allocate the labels {2n + 2, 2n + 3, . . . , . . . 3n + 1} to the vertices v1, v2, . . . , vn in
any order.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Fln⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.
Further
f(u) | f(uj), f(u) | f(wj), 1 ≤ j ≤ n
and
f(w1) | f(vk) whenever k is odd, 1 ≤ k ≤ n.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Fln ⊕K1,n is DC graph.
Example 3.3.6. DC labeling in Fl4 ⊕K1,4 is demonstrated in the following Figure
3.7.
27
3.3. Some New DC Graphs With the Use of Ringsum Operation
u1
u2
u3
u4
v1
v2 v3
v4
u1
3
7
954
6
8
2
10
11 12
13
w2
w1
w3
w4
Figure 3.7: DC labeling in Fl4 ⊕K1,4
Theorem 3.3.7. Sn ⊕K1,n is DC graph.
Proof. Let V (Sn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Sn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Sn⊕K1,n) = {ujuj+1, unu1 | 1 ≤ j ≤ n−1}⋃{u1uj | 3 ≤ j ≤ n−1}⋃{u1vj |1 ≤ j ≤ n}.It is to be noted that, |V (Sn ⊕K1,n)| = 2n and |E(Sn ⊕K1,n)| = 3n− 3.
Consider a bijection f : V (Sn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 2.
For the vertices u2, u3, . . . , uk, allocate the vertex labels as per the below ordered
pattern upto it generate k edges with label 1.
1, 2× 21, 2× 22 . . . , 2× 2k1 ,
3, 3× 21, 3× 22 . . . , 3× 2k2 ,
5, 5× 21, 5× 22 . . . , 5× 2k3 ,
. . . , . . . , . . . , . . . ,
. . . , . . . , . . . , . . .
Observe that (2m− 1)2km ≤ n and km ≥ 0 (m ≥ 1).
(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.
28
3.3. Some New DC Graphs With the Use of Ringsum Operation
Then for remaining vertices of uk+1, uk+2, . . . , un, allocate the vertex labels such that
the consecutive vertices do not generate edge label 1.
f(vj) = f(un) + j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge condition
n ≡ 0, 2(mod 4) ef (0) =⌈3n−1
2
⌉, ef (1) =
⌊3n−1
2
⌋
n ≡ 1, 3(mod 4) ef (1) = 3n−12
= ef (0)
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Sn ⊕K1,n is DC graph.
Example 3.3.7. DC labeling in the graph S7⊕K1,7 is demonstrated in the following
Figure 3.8.
u3
u4u5
u6
u7
u1
u2
v3 v4 v5 v6 v7v1 v2
5
9 13
6
10 12 1411
1
2
7
8
3
4
Figure 3.8: DC labeling in S7 ⊕K1,7
Theorem 3.3.8. Pn ⊕K1,n is DC graph.
Proof. Let V (Pn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Pn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Pn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Pn ⊕K1,n)| = 2n and |E(Pn ⊕K1,n)| = 2n− 1.
29
3.3. Some New DC Graphs With the Use of Ringsum Operation
Consider a bijection f : V (Pn ⊕K1,n)→ {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 2.
f(v1) = 1.
f(uj) = 2j − 1; 2 ≤ j ≤ n.
f(vk) = 2k; 2 ≤ k ≤ n.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(Pn⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.
Also
f(u1) | f(vk) 1 ≤ k ≤ n
.
By looking into the above prescribed pattern,
ef (1) = n, ef (0) = n− 1.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, Pn ⊕K1,n is DC graph.
Example 3.3.8. DC labeling in graph P5 ⊕ K1,5 is demonstrated in the following
Figure 3.9.
u1u2u3u4u5
v1
v2
v3
v4
v5
3 2579
14
6
8
10
Figure 3.9: DC labeling in P5 ⊕K1,5
Theorem 3.3.9. DFn ⊕K1,n is DC graph.
Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.
30
3.3. Some New DC Graphs With the Use of Ringsum Operation
Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(DFn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uvj | 1 ≤ j ≤ n}⋃{uuj | 1 ≤ j ≤n}⋃{wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn ⊕K1,n)| = 2n+ 2 and |E(DFn ⊕K1,n)| = 4n− 1.
Consider a bijection f : V (DFn ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} defined as below.
f(u) = 1.
f(w) = p, where p is highest prime number ≤ 2n+ 2.
f(v1) = 2.
f(u1) = 3.
f(uj) = 2j; 2 ≤ j ≤ n.
f(vk) = 2k + 1; 2 ≤ k ≤ n− 1.
f(vn) = 2n+ 2.
As per this pattern, label the vertices such that, for any edge ujuj+1 ∈ E(DFn ⊕K1,n),
f(uj) - f(uj+1), 1 ≤ j ≤ n− 1.
Also
f(u) | f(vk), 1 ≤ k ≤ n
and
f(u) | f(uj), f(w) - f(uj) 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n, ef (0) = 2n− 1.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, DFn ⊕K1,n is DC graph.
Example 3.3.9. DC labeling in graph DF5 ⊕K1,5 is demonstrated in the following
Figure 3.10.
31
3.3. Some New DC Graphs With the Use of Ringsum Operation
u1 u2 u3 u4 u5
79
v1
v2
v3
v4
v5
u
w
1
3 4 6 8 10
5
12
11
2
Figure 3.10: DC labeling in DF5 ⊕K1,5
Theorem 3.3.10. K2,n ⊕K1,n is DC graph.
Proof. Let V (K2,n ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (K2,n) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.
Consider a bijection f : V (K2,n ⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} defined as below.
f(u) = 1.
f(w) = p,where, p = max {x | x is the largest prime number x ≤ 2n+ 2}.
f(uj) = j + 1; 1 ≤ j ≤ n.
Allocate the labels {n+2, n+3, . . . , p−1, p+1, . . . 2n+2} to the vertices v1, v2, . . . , vn
in any order.
As per this pattern, label the vertices such that, for any edge e ∈ E(K2,n ⊕K1,n),
f(u) | f(uj), f(w) - f(uj) 1 ≤ j ≤ n.
Also
f(u1) | f(vk) whenever k is even, 1 ≤ k ≤ n.
As per the above stated labeling pattern,
32
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n2
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌈3n2
⌉, ef (0) =
⌊3n2
⌋
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, K2,n ⊕K1,n is DC graph.
Example 3.3.10. DC labeling in K2,7⊕K1,7 is demonstrated in the following Figure
3.11
u1
u2
v1
v2
u3
v3 v4
v5
u4
u
u5
u6
u7
v6
v7
1
2
13
5
7
9
3
4
6
8
16
15
141211
10
w
Figure 3.11: DC labeling in K2,7 ⊕K1,7
Remark 3.3.2. In each of the above theorems, for the ringsum operation with K1,n,
one can consider any arbitrary vertex of the graph G under consideration and by
different permutations of the vertex labels provided in the respective labeling pattern,
one can easily check that the resultant graph still admit DC labeling.
3.4 DC Labeling With the Use of Switching Invariance in
Cycle Allied Graphs
Vaidya and Shah[49] derived some captivating results on DC labeling of the graphs
constructed from switching a vertex in different graphs. In the current section we
demonstrate few graphs constructed from switching invariance in cycle allied graphs
for DC labeling.
33
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
Theorem 3.4.1. Gv is DC, where G is gear Gn graph and v is not apex vertex.
Proof. Let V (Gn) = {uj | 0 ≤ j ≤ 2n}, where u0 is the apex vertex and uj(1 ≤ j ≤2n) are vertices of gear graph Gn such that
deg(ui) =
2; if j is even.
3; if j is odd.
Let E(Gn) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 1 ≤ j ≤ 2n− 1}⋃{u2nu1}.(Gn)ui
∼= (Gn)uj , where d(ui) = d(uj).
Let (Gn)uj denote the graph constructed from switching of vertex uj (j = 1, 2) of
Gn.
Corresponding to the vertices of different degree in Gn, it is required to discuss
following two cases.
Case 1: d(u1) = 3.
Then by the effect of switching operation, the edge set of (Gn)u1 is
E((Gn)u1) = {u0u2j−1 | 2 ≤ j ≤ n}⋃{ujuj+1 | 2 ≤ j ≤ 2n − 1}⋃{u1uj | 3 ≤ j ≤2n− 1}.Here note that, |V ((Gn)u1)| = 2n+ 1 and |E((Gn)u1)| = 5n− 6.
Consider a bijection f : V ((Gn)u1)→ {1, 2, 3, . . . , 2n+ 1} defined as below.
Our aim is to generate b5n−62c edges with label 1 and d5n−6
2e edges with label 0.
Let f(u0) = p, where p is the largest prime, p ≤ 2n+ 1.
Then n− 1 edges with label 0 will be generated.
Assign f(u1) = 1, which generates 2n− 3 edges with label 1.
Now it remains to generate k = b5n−62c − (2n− 3) edges with label 1.
For the vertices u2, u3, . . . , u2n, assign the vertex labels as per the following ordered
pattern upto it generate k edges with label 1.
2, 2× 21, 2× 22 . . . , 2× 2k1 ,
3, 3× 21, 3× 22 . . . , 3× 2k2 ,
5, 5× 21, 5× 22 . . . , 5× 2k3 ,
. . . , . . . , . . . , . . . ,
34
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
. . . , . . . , . . . , . . .
Observe that (2m− 1)2km ≤ n and km ≥ 0 (m ≥ 1).
(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.
Then for remaining vertices of (Gn)u1 , assign the vertex labels such that the consec-
utive vertices do not generate edge label 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 5n−62
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 5n−52, ef (0) = 5n−7
2
Then we get, |ef (1)− ef (0)| ≤ 1 in this case.
Case 2: d(u2) = 2
Then by the effect of switching operation, the edge set of (Gn)u2 is
E((Gn)u2) = {u0u2i−1 | 1 ≤ i ≤ n}⋃{uiui+1 | 3 ≤ i ≤ 2n − 1}⋃{u2nu1}⋃{u2ui |
4 ≤ i ≤ 2n}.Here note that, |V (Gn))u2| = 2n+ 1 and |E(Gn))u2| = 5n− 4.
Using the same labeling pattern as in Case 1, we get the following.
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 5n−42
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 5n−32, ef (0) = 5n−5
2
Then we get, |ef (1)− ef (0)| ≤ 1 in this case.
That is, (Gn)v is DC, v is not apex vertex.
Example 3.4.1. The following Figure 3.12 demonstrates
(i) Gear graph G6.
(ii) DC labeling in (G6)v, where d(v) = 3.
(iii) DC labeling in (G6)v, where d(v) = 2.
35
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
u0
u2
u3
u4
u5
u6
u7
u8
u9
u10
u12
u1
u11
13
7
1
5
9
2
3
4
6
8 10
11
122
37
4
6
9
10
8
1112
13
1
5
Figure 3.12: DC labeling in (G6)v
Theorem 3.4.2. Gv is DC, where G is shell graph Sn and v is not apex vertex.
Proof. Let V (Sn) = {uj | 0 ≤ j ≤ n − 1}, where u0 is the apex vertex and uj(1 ≤j ≤ n− 1) are vertices of shell graph Sn, where
deg(uj) =
2; if j = 1 and n− 1.
3; if j = 2, 3, . . . , n− 2.
Let E(Sn) = {u0uj | 2 ≤ j ≤ n− 2}⋃{ujuj+1 | 0 ≤ j ≤ n− 2}⋃{u0un−1}.(Sn)ui
∼= (Sn)uj , where d(ui) = d(uj).
Let (Sn)uj denote the graph constructed from switching of vertex uj (j = 1, 2) of
Sn.
Corresponding to the vertices of different degree in Sn, it is required to discuss fol-
lowing two cases.
Case 1: d(u2) = 3.
Then by the effect of switching operation, the edge set of (Sn)u2 is
E((Sn)u2) = {u0uj | 3 ≤ j ≤ n−2}⋃{ujuj+1 | 3 ≤ j ≤ n−2}⋃{un−1u0}{u0u1}⋃{u2uj |
4 ≤ j ≤ n− 1}.Here note that, |V ((Sn)u2)| = n and |E((Sn)u2)| = 3n− 10.
Consider a bijection f : V ((Sn))u2)→ {1, 2, 3, . . . , n} defined as below.
Subcase 1: n = 6.
f(uj) = j + 1; j = 0, 1, 2.
f(uj) = j; j = 4.
36
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
f(u3) = p, where p = max {x | x is a prime number and x ≤ n}.
f(un−1) = n.
Subcase 2: n 6= 6, n ∈ N.
Label the vertices u0, u1, . . . un−1 as per the following pattern.
1, 1× 21, 1× 22 . . . , 1× 2k1 ,
3, 3× 21, 3× 22 . . . , 3× 2k2 ,
5, 5× 21, 5× 22 . . . , 5× 2k3 ,
. . . , . . . , . . . , . . . ,
. . . , . . . , . . . , . . .
Observe that (2m− 1)2km ≤ n and km ≥ 0 (m ≥ 1).
(2m− 1)2α | (2m− 1)2α+1 and (2m− 1)2ki - 2m+ 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−102
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 3n−92, ef (0) = 3n−9
2
Then we get, |ef (1)− ef (0)| ≤ 1 in this case.
Case 2: d(u1) = 2.
Then by the effect of switching operation, the edge set of (Sn)u1 is
E((Sn)u1) = {u0uj | 2 ≤ j ≤ n − 1}⋃{ujuj+1 | 2 ≤ j ≤ n − 1}⋃{un−1u0}{u1uj |3 ≤ j ≤ n− 1}.Here note that, |V ((Sn)u1)| = n and |E((Sn)u1)| = 3n− 8.
Let us f : V ((Sn)u1)→ {1, 2, 3, . . . , n} defined as below
f(uj) = j + 1; 0 ≤ j ≤ n− 1.
By looking into the above prescribed pattern,
37
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−82
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 3n−72, ef (0) = 3n−9
2
Then we get, |ef (1)− ef (0)| ≤ 1 in this case.
That is, (Sn)v is DC, v is not apex vertex.
Example 3.4.2. The following Figure 3.13 demonstrates
(i) Shell graph S7.
(ii) DC labeling in (S7)v, where d(v) = 3.
(iii) DC labeling in (S7)v, where d(v) = 2.
2
1
4
3
5
6
7
u2
u3 u4
u5
u1
u0
u6
2
1
3
4 5
6
7
Figure 3.13: DC labeling in (S7)v
Theorem 3.4.3. Gv is DC, where G is flower graph Fln and v is not apex vertex.
Proof. Let V (Fln) = {uj | 0 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where d(u0) = 2n,
d(uj) = 4(1 ≤ j ≤ n) are the internal vertices and d(vj) = 2(1 ≤ j ≤ n) are the
external vertices.
Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 1 ≤ j ≤ n}.(Fln)ui
∼= (Fln)uj , where d(ui) = d(uj).
Let (Fln)u1 and (Fln)v1 denote the graph constructed from switching of vertex u1
and v1 of Fln respectively.
Corresponding to the vertices of different degree in Fln, it is required to discuss
following two cases.
38
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
Case 1: d(u1) = 4.
Then by the effect of switching operation, the edge set of (Fln)u1 is
E((Fln)u1) = {ujuj+1 | 2 ≤ j ≤ n}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.Here note that, |V ((Fln)u1)| = 2n+ 1 and |E((Fln)u1)| = 6n− 8.
Consider a bijection f : V ((Fln)u1)→ {1, 2, 3, . . . , 2n+ 1} defined as below.
f(u0) = 1.
f(uj) = 2j; 1 ≤ j ≤ n.
f(vk) = 2k + 1; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 4 = ef (0).
Then we get, |ef (1)− ef (0)| ≤ 1, in this case.
Case 2: d(v1) = 2.
Then by the effect of switching operation, the edge set of (Fln)v1 is
E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ i ≤ n}⋃{v1uj | 2 ≤ j ≤ n}⋃{v1vj | 2 ≤ j ≤ n}.Also it is to be noted that, |V ((Fln)v1)| = 2n+ 1 and |E((Fln)v1)| = 6n− 4.
Let us a function f from ((Fln)v1) to {1, 2, 3, . . . , 2n+ 1} defined as below.
f(u0) = 1.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
f(vk) = 2k; 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 2 = ef (0).
Then we get, |ef (1)− ef (0)| ≤ 1, in this case.
That is, (Fln)v is DC, v is not apex vertex.
39
3.4. DC Labeling With the Use of Switching Invariance in Cycle Allied Graphs
Example 3.4.3. The following Figure 3.14 demonstrates
(i) Flower graph Fl4.
(ii) DC labeling in (Fl4)v1, where d(v1) = 2.
(iii) DC labeling in (Fl4)u1, where d(u1) = 4.
6
2
1
8
9
7
3
4
5
v1v4
u1 u4
v2v3
u2 u3
u0
3
4 6
8
9
75
1
2
Figure 3.14: DC labeling in (Fl4)v
Remark 3.4.1. The graph constructed from switching of a pendant vertex in star
K1,n is isomorphic to K2,n−1 and hence confess DC labeling (Refer [44]).
Remark 3.4.2. (K2,n)v is DC graph because of the following two possibilities.
1. Switching of a vertex with degree 2 in K2,n is isomorphic to K3,n−1 and confess
DC labeling (Refer [44]).
2. Switching of a vertex with degree n in K2,n is isomorphic to K1,n+1 and hence
confess DC labeling (Refer [44]).
Remark 3.4.3. (K3,n)v is DC graph because of the following two possibilities.
1. Switching of a vertex with degree three in K3,n is isomorphic to K4,n−1 and
hence confess DC labeling (Refer [44]).
2. Switching of a vertex with degree n in K3,n is isomorphic to K2,n+1 and hence
confess DC labeling (Refer [44]).
40
3.5. DC Labeling With the Use of Duplication of a Vertex/Edge in Star Graph
3.5 DC Labeling With the Use of Duplication of a Ver-
tex/Edge in Star Graph
Vaidya and Prajapati[55] derived some attractive results on prime labeling of graphs
constructed from duplication of graph elements. In this section we demonstrate some
DC graphs constructed from duplication of vertex/edge in star K1,n.
Theorem 3.5.1. The graph constructed from duplication of a vertex in star K1,n is
DC.
Proof. Let V (K1,n) = {v0, vj | 1 ≤ j ≤ n}, where v0 is the apex vertex and
d(vj) = 1(1 ≤ j ≤ n). E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplication of any vertex vj by
vertex v′j in K1,n.
Corresponding to the vertices of different degree in K1,n, it is required to discuss
following two cases.
Case 1: d(v0) = n.
The graph constructed from duplication of apex vertex v0 in K1,n is the graph K2,n
and hence it is DC graph (Refer [44]).
Case 2: d(v1) = 1.
The graph constructed from duplication of any pendant vertex in K1,n is a star
graph K1,n+1 and hence it is DC graph (Refer [44]).
Thus in each case the proof is an immidiate outcome of the result proved by
Varatharajan, Navanaeethakrishnan and Nagarajan [44].
Theorem 3.5.2. The graph constructed from duplication of a vertex by an edge in
star K1,n is DC.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj are the
pendant vertices, j = 1, 2, . . . , n.
Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplicating an arbitrary vertex
vj by an edge v′jv′′j in K1,n.
41
3.5. DC Labeling With the Use of Duplication of a Vertex/Edge in Star Graph
It is to be noted that, |V (G)| = n+ 3 = |E(G)|.Corresponding to the vertices of different degree in K1,n, it is required to discuss
below two possibilities.
Case 1: d(v0) = n.
Let us consider the vertex be v0 and it’s duplicated edge be v′0v′′0 .
.Let us a function f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.
f(v0) = 2.
f(v1) = 1.
f(v′0) = n+ 2.
f(v′′0) = n+ 3.
f(vj) = j + 1; 2 ≤ j ≤ n.
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(vj) = 1.
Let us consider the vertex be vj and it’s duplicated edge be v′jv′′j .
WLOG we may assume that vj = vn.
Let us a function f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.
f(v0) = 2.
f(v′n) = n+ 3.
f(v′′n) = 1.
f(vj) = j + 2; 1 ≤ j ≤ n.
Then we get, |ef (1)− ef (0)| ≤ 1 .
That is, the graph constructed from duplicating an arbitrary vertex by an edge in
K1,n is DC.
Example 3.5.1. DC labeling of the graphs constructed from duplication of apex ver-
tex v0 by an edge v′0v′′0 and pendant vertex v5 by an edge v′5v
′′5 in K1,5 are demonstrated
in the following Figure 3.15.
42
3.5. DC Labeling With the Use of Duplication of a Vertex/Edge in Star Graph
v5
1
2
3
5
6 7
8
4
v0
v1
v2
v3 v4
v5v0
v1
v2
v3 v4
v5'
1
87v5
v5"
65
3
4
2
v0
v0'
v0"
v1
v2
v3
v4
Figure 3.15: DC labeling in the graph constructed from duplication of vertex by edge in K1,5
Corollary 3.5.1. The graph constructed from duplication of an edge by a vertex in
star K1,n is DC.
Proof. The graph constructed from duplicating an edge by a vertex in star K1,n is
same as the graph constructed from duplicating apex vertex by an edge; which is a
DC graph (Refer Theorem 3.5.2).
Theorem 3.5.3. The graph constructed from duplication of an edge in star K1,n is
DC.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj are the
pendant vertices, j = 1, 2, . . . , n.
Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplicating an edge v0vn by a
new edge v′0v′n in K1,n with edge set E(G) = {v0vj | 1 ≤ j ≤ n}⋃{v′0vj | 1 ≤ j ≤
n}⋃{v′0v′n}.Here note that, |V (G)| = n+ 3 and |E(G)| = 2n.
Consider a bijection f : V (G)→ {1, 2, . . . , n+ 3} defined as below.
f(v0) = 1.
f(v′0) = p,where p = max {x | x is a prime number and x ≤ n+ 4}.
Allocate the labels {2, 3, . . . , p− 1, p+ 1, . . . , n+ 3} to the vertices v1, v2, . . . , vn, v′n
in any order.
Then we get, |ef (0)− ef (1)| ≤ 1.
43
3.6. Conclusion and Scope for Further Research
That is, the graph constructed from duplicating any edge in K1,n is DC.
Example 3.5.2. DC labeling of the graph constructed from duplication of an edge
v0v8 by edge v′0v′8 in K1,8 is demonstrated in the following Figure 3.16.
1 9v0
v1
v2
v3
v4 v5
v6
v8
v7
v0 v8
v7
v1
v2
v3
v4
v5
v6
v'0v'8
2
1110
7
8
5
6
3
4
Figure 3.16: DC labeling in the graph constructed from duplication of edge v0v8 in K1,8
3.6 Conclusion and Scope for Further Research
In this chapter we have emanated DC labeling for larger graphs constructed from the
standard graphs by means of various graph operations such as ringsum of different
graphs with star graph, switching of vertex in cycle allied graphs and duplication of
vertex in star allied graphs. At the end, we pose some open problems listed below.
Problem 3.6.1. Derive essential and adequate condition (if any) for any graph to
be DC graph.
Problem 3.6.2. To investigate some new DC graphs with respect to other graph
operations.
Problem 3.6.3. Classify/Generalize the graphs G such that G⊕K1,n is DC graph.
(Here |V (G)| may or may not be equal to n.)
The next chapter is intended to discuss square DC, cube DC and vertex odd DC
labeling of graphs.
44
CHAPTER 4
Square Divisor Cordial, Cube
Divisor Cordial and Vertex Odd
Divisor Cordial Labeling of
Graphs
4.1 Introduction
Divisor cordial labeling has been introduced in the earlier chapter while the current
chapter aims to give a transitory collection of few labelings having DC theme.
4.2 Square DC Labeling
4.2.1 Introduction
Inspired by the idea of DC labeling, Murugesan et al.[56] have established a variant of
DC labeling namely square DC labeling of graphs. In a DC graph with DC labeling
f , the edge label for an edge ab is produced by using the condition “whether f(a)
divides f(b) or not”; while in the square DC labeling, the edge label for an edge ab
is produced by using the condition “whether (f(a))2 divides f(b) or not”. Thus the
concept of square DC labeling differs from DC labeling merely by replacing label of
45
4.2. Square DC Labeling
one of the end vertices of the centain edge by its square to produce required edge
label.
Definition 4.2.1 (Murugesan et al.[56]). Let G = (V,E) be a simple graph with
order p and size q. Consider a bijection f : V (G) → {1, 2, . . . , |V (G)|} and let the
induced function f ∗ : E(G)→ {0, 1} be defined as
f ∗(uv) =
1; if [f(u)]2 | f(v) or [f(v)]2 | f(u)
0; otherwise
Then the function f is called a square DC labeling if |ef (0)− ef (1)| ≤ 1.
A graph which confesses square DC labeling is called square DC graph.
Example 4.2.1. A square DC labeling in shell graph S7 is demonstrated in the
following Figure 4.1.
v2
v3 v4
v5
v0
v6
1
2
4
3
5
6
7v1
Figure 4.1: Square DC labeling in S7
It is easy to observe that a graph may admit more than one square DC labeling.
However, if one such labeling exists, then the graph becomes square DC graph.
4.2.2 Some Known Results on Square DC Labeling
In first paper on square DC labeling[56], Murugesan et al. have derived square DC
labeling for some basic graphs and derived the following results.
Theorem 4.2.1. The path Pn is square DC iff n ≤ 12.
Theorem 4.2.2. The cycle Cn is square DC iff 3 ≤ n ≤ 11.
46
4.2. Square DC Labeling
Theorem 4.2.3. The wheel graph Wn is square DC.
Theorem 4.2.4. The graph K1,n is square DC iff n = 2, 3, 4, 5 or 7.
Theorem 4.2.5. The graph K2,n is square DC.
Theorem 4.2.6. The graph K3,n is square DC iff n = 1, 2, 3, 5, 6, 7 or 9.
Theorem 4.2.7. The complete graph Kn is square DC iff n = 1, 2, 3 or 5.
Vaidya and Shah[51] have derived some benchmark results on square DC labeling.
Few such results are stated below.
Theorem 4.2.8. The flower graph Fln and bistar Bn,n are square DC.
Theorem 4.2.9. Restricted B2n,n graph and D2(Bn,n) are square DC.
Theorem 4.2.10. Splitting graph of star K1,n and Bistar Bn,n are square DC.
Theorem 4.2.11. Degree splitting graph of bistar Bn,n and path Pn are square DC.
Kanani and Bosmia[28] have discussed square DC labeling for the following graphs
using the graph operation “switching of a vertex”.
Theorem 4.2.12. The graph constructed from switching of a vertex in the bistar
Bm,n and the comb graph Pn �K1 are square DC.
Theorem 4.2.13. (Cn �K1)v and (ACn �K1)v are square DC.
Theorem 4.2.14. The graph constructed from switching of a vertex except apex
vertex in the helm Hn and the gear graph Gn are square DC.
However, there is no such standard relation between either of the two labelings; like,
we may find a graph which admits one labeling but not the other. To observe this
matter more effectively, the list of graph families satisfying/ not satisfying certain
labeling is shown below.
• The wheel graph Wn admits both DC and square DC labeling (Refer [44] and
[56]).
• The path graph P13 is DC graph but not square DC (Refer [44] and [56]).
• The complete graph K7 is neirher DC nor square DC graph (Refer [44] and
[56]).
47
4.3. Cube DC Labeling
4.3 Cube DC Labeling
4.3.1 Introduction
Motivated by two concepts, DC labeling and square DC labeling, Kanani and
Bosmia[25] have established another variant of DC labeling namely cube DC la-
beling. Cube DC labeling can be considered as an extension of square DC labeling
by considering cube of label of one of the end vertex of a certain edge.
Definition 4.3.1 (Kanani and Bosmia[25]). Let G = (V,E) be a simple graph with
order p and size q. Consider a bijection f : V (G) → {1, 2, . . . , |V (G)|} and let the
induced function f ∗ : E(G)→ {0, 1} be defined as
f ∗(uv) =
1; if [f(u)]3 | f(v) or [f(v)]3 | f(u)
0; otherwise
Then the function f is called a CDC labeling if |ef (0)− ef (1)| ≤ 1.
A graph which confesses CDC labeling is called CDC graph.
Example 4.3.1. A CDC labeling in graph K2,7 is demonstrated in the following
Figure 4.2.
u1 u2 u3 u4 u5 u6 u7
2 5 73 4 6 8
u1 13
w
Figure 4.2: CDC labeling in K2,7
CDC labeling in a graph (if exists) may not be unique.
48
4.4. Vertex Odd DC Labeling
4.3.2 Some Known Results on Cube DC Labeling
In two different research papers on CDC labeling[25, 26], Kanani and Bosmia have
established CDC labeling for some basic graphs and derived the following attractive
results.
Theorem 4.3.1. The path Pn is CDC graph iff n ≤ 6, n = 8.
Theorem 4.3.2. The cycle Cn is CDC graph iff n = 3, 4, 5.
Theorem 4.3.3. The wheel Wn and flower graph Fln are CDC graphs.
Theorem 4.3.4. The fan graph Fn is CDC graph for all n.
Theorem 4.3.5. Kn is CDC graph iff n ≤ 4.
Theorem 4.3.6. The star graph K1,n is a CDC graph if n ≤ 3.
Theorem 4.3.7. The graph K2,n is a CDC graph.
Theorem 4.3.8. The graph K3,n is CDC if n = 1, 2.
Theorem 4.3.9. The bistar Bn,n and restricted B2n,n are CDC graphs.
However, there is no such standard relation between either of the two square DC
and CDC labelings; like, we may find a graph which admits one labeling but not the
other. To observe this matter more effectively, the list of graph families satisfying/
not satisfying certain labeling is shown below.
• The wheel graph Wn admits both square DC and CDC (Refer [56] and [25]).
• The graph K3,7 is square DC but it is not CDC (Refer [56] and [26]).
• The complete graph K4 is not square DC but it is CDC (Refer [56] and [26]).
• The star graph K1,n(n ≥ 8) is neirher square DC nor CDC (Refer [56] and[26]).
4.4 Vertex Odd DC Labeling
4.4.1 Introduction
Inspired by the idea of DC labeling and odd labeling, Muthaiyan and Pugalenthi[3]
introduced a special type of DC labeling called vertex odd DC labeling.
49
4.4. Vertex Odd DC Labeling
Definition 4.4.1 (Muthaiyan and Pugalenthi[3]). Let G = (V,E) be a simple graph
with order p and size q. Consider a bijection f : V (G) → {1, 3, . . . , 2|V (G)| − 1}and let the edge labeling function f ∗ : E(G)→ {0, 1} be defined as
f ∗(uv) =
1; if f(u) | f(v) or f(v) | f(u)
0; otherwise
Then the function f is called a VODC labeling if |ef (0)− ef (1)| ≤ 1.
A graph which confesses VODC labeling is called VODC graph.
In any VODC graph, VODC labeling may or may not be unique.
Example 4.4.1. A VODC labeling in flower graph Fl7 is demonstrated in the fol-
lowing Figure 4.3.
v0
v1v2
v3
v4
v5
v6
v7
u1
u2
u3
u4
u5
u6
u7
1
3
15
9
5
7
1113
17
19
21
27
29
2325
Figure 4.3: VODC labeling in Fl7
4.4.2 Some Known Results on VODC Labeling
In two different research papers ([3], [4]), Muthaiyan and and Pugalenthi have proved
various graphs to be VODC graphs. These results are stated below.
Theorem 4.4.1. K2,n is VODC.
Theorem 4.4.2. Shell graph Sn is VODC.
50
4.4. Vertex Odd DC Labeling
Theorem 4.4.3. The wheel graph Wn and bistar Bn,n are VODC graphs.
Theorem 4.4.4. The helm Hn and the flower graph Fln are VODC graphs.
Theorem 4.4.5. (Hn)v is a VODC graph, where v is apex vertex.
Theorem 4.4.6. (Pn)v is VODC graph, where v is pendant vertex.
Theorem 4.4.7. (Cn)v is a VODC graph.
Theorem 4.4.8. Barycentric subdivision of K1,n is VODC graph.
Theorem 4.4.9. Restricted square graph, Splitting graph and degree splitting graph
of Bn,n are VODC graph.
Sugumaran and Suresh[48] have discussed further results on VODC labeling of
graphs, few of them are listed below.
Theorem 4.4.10. The gear graph Gn is VODC.
Theorem 4.4.11. The graph P2 +mK1 is a VODC graph.
Theorem 4.4.12. The 1-weak shell graph C(n, n−3) and 2-weak shell graph C(n, n−4) are VODC graphs.
However, there is no such standard relation between either of the two CDC and
VODC labelings; like, we may find a graph which admits one labeling but not the
other. To observe this matter more effectively, the list of graph families satisfying/
not satisfying certain labeling is shown below.
• The wheel graph Wn is both CDC and VODC (Refer [25] and [3]).
• The complete graph K4 is CDC (Refer [26]) but it is not VODC (easy to check).
• The graph W7⊕K1,7 is not CDC (easy to check) but it is VODC (Refer theorem
4.8.5).
• The star graph K1,6 is neirher CDC (Refer [25]) nor VODC (easy to check).
51
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
4.5 New Results on Square DC, Cube DC and Vertex Odd
DC Labeling of Graphs
4.5.1 Introduction
In this section we derive new graphs admitting square DC, CDC and VODC labeling.
Definition 4.5.1 ([18]). A graph G = (V,E) is called a tripartite graph if vertex set
V (G) can be divided into three nonempty disjoint subsets V1, V2 and V3 such that
vertices in the same set are not adjacent to each other.
The complete tripartite graph with |V1| = n1, |V2| = n2, |V3| = n3 is denoted by
Kn1,n2,n3.
Theorem 4.5.1. K1,1,n is square DC graph.
Proof. Let V (K1,1,n) = {u, v, uj | 1 ≤ j ≤ n}, where d(u) = n + 1 = d(v) and
d(uj) = 2(1 ≤ j ≤ n).
Let E(K1,1,n) = {uv, uuj, vuj | 1 ≤ j ≤ n}.It is to be noted that, |V (K1,1,n)| = n+ 2 and |E(K1,1,n)| = 2n+ 1.
Consider a bijection f : V (K1,1,n)→ {1, 2, 3, . . . , n+ 2} defined as below.
f(u) = 1.
f(v) = p,where p = max {x | x is a prime number and x ≤ n+ 2}.
Allocate labels {2, 3, . . . , p − 1, p + 1, . . . , n + 2} to the vertices uj(1 ≤ j ≤ n) of
K1,1,n in any order.
As per this pattern, the vertices are labeled such that
[f(u)]2 | f(uj), 1 ≤ j ≤ n.
Also as f(v) is prime,
[f(v)]2 - f(uj), 1 ≤ j ≤ n.
Further since f(u) = 1,
[f(u)]2 | f(v).
52
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
By looking into the above prescribed pattern,
ef (1) = ef (0) + 1.
i.e. |ef (0)− ef (1)| ≤ 1.
Therefore the graph under consideration admits square DC labeling and hence K1,1,n
is a square DC graph.
Example 4.5.1. Square DC labeling for the special case of n = 6 in above theorem
is demonstrated in the following Figure 4.4.
1 7u v
u1
u2
u3
u4
u5
u6
6
4
2
3
5
8
Figure 4.4: Square DC labeling in K1,1,6
Corollary 4.5.1. K1,1,n is a CDC graph.
Proof. The labeling function can be defined same as in Theorem 4.5.1. Using the
same arguments considered in above theorem, we have
[f(u)]3 | f(uj), 1 ≤ j ≤ n.
Also as f(v) is prime,
[f(v)]3 - f(uj), 1 ≤ j ≤ n.
Further as f(u) = 1,
[f(u)]3 | f(v).
Hence, K1,1,n is a CDC graph.
Corollary 4.5.2. K1,1,n is a VODC graph.
53
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Proof. By considering the similar labeling function as defined in Theorem 4.5.1 and
also using almost same arguments, one can conclude that K1,1,n is a VODC graph.
Corollary 4.5.3. K2 + nK1 is a square DC, CDC and VODC graph.
It is to be noted that: K1,1,n∼= K2 + nK1
Definition 4.5.2 ([18]). Umbrella graph U(m,n) (m > 2 and n > 1) is the graph
constructed from appending a path Pn to the apex of a fan Fm = Pm +K1
Theorem 4.5.2. Umbrella U(n, 3) is a square DC graph.
Proof. Let V (U(n, 3)) = {uj, v1, v2, v3 | 1 ≤ j ≤ n}, where d(v1) = n+ 1, d(v2) = 2,
and d(v3) = 1 and uj(2 ≤ j ≤ n) are the vertices of the path of Fn with v1 as a
central vertex of Fn.
Let E(U(n, 3)) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{v1uj, | 1 ≤ j ≤ n}⋃{v1v2, v2v3}.It is to be noted that, |V (U(n, 3))| = n+ 3 and |E(U(n, 3))| = 2n+ 1.
Consider a bijection f : V (U(n, 3))→ {1, 2, 3, . . . , n+ 3} as per the following.
f(v1) = 1.
f(uj) = j + 1; 1 ≤ j ≤ n.
f(v2) = n+ 2.
f(v3) = n+ 3.
By looking into the above prescribed pattern,
ef (1) = n+ 1, ef (0) = n.
Hence
ef (1) = ef (0) + 1.
Then in each case, we get |ef (0)− ef (1)| ≤ 1.
Therefore umbrella graph U(n, 3) is a square DC graph.
Example 4.5.2. Square DC labeling of the graph U(9, 3)is demonstrated in the
following Figure 4.5.
54
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
u1
u2
u3
u4
u5
u8
u9
u6
u7
v1
v2
v3
1
4
2
3
65 7
8
9
10
11
12
Figure 4.5: Square DC labeling in U(9, 3)
Corollary 4.5.4. Umbrella U(n, 3) is a CDC graph.
Corollary 4.5.5. Umbrella U(n, 3) is a VODC graph.
Definition 4.5.3. [18] A one point union C(t)n of t copies of cycles Cn is the graph
constructed by taking v as a common vertex such that any two cycles C(i)n and
C(j)n (i 6= j) are edge disjoint and do not have any vertex in common except v.
The one point union of t (≥ 1) cycles, each of length n is denoted by C(t)n .
Theorem 4.5.3. C(t)4 is a square DC graph.
Proof. Let V (C(t)4 ) = {vi,j | 1 ≤ i ≤ t, 1 ≤ j ≤ 4}, where v1,1 = v2,1 = v3,1 = v4,1 = v
(say); i.e. v is a common vertex.
Let E(C(t)4 ) = {vi,jvi,j+1 | 1 ≤ i ≤ t, 1 ≤ j ≤ 3}⋃{vi,nvi,1 | 1 ≤ i ≤ t}.
It is to be noted that, |V (C(t)4 )| = 3t+ 1 and |E(C
(t)4 )| = 4t.
Consider a bijection f : V (C(t)4 )→ {1, 2, 3, . . . , 3t+ 1} is defined as below.
f(v) = 1.
f(vi,2) = 3i− 1; 1 ≤ i ≤ t.
f(vi,3) = 3i; 1 ≤ i ≤ t.
f(vi,4) = 3i+ 1; 1 ≤ i ≤ t.
By looking into the above prescribed pattern,
ef (1) = 2t = ef (0).
55
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, C(t)4 is a square DC graph.
Example 4.5.3. Square DC labeling in C(5)4 is demonstrated in the following Figure
4.6.
1
42
3
10
9
8
12
11
13
5
6 7 14 15
16
Figure 4.6: Square DC labeling in C(5)4
Corollary 4.5.6. C(t)4 is a CDC graph.
Corollary 4.5.7. C(t)4 is a VODC graph.
Definition 4.5.4 ([37]). Consider k copies of graph G (wheel, star, fan and friend-
ship) say G(1), G(2), . . . , G(k). Then, the graph < G(1), G(2) . . . , G(t) > is constructed
by joining apex vertex of each G(i) and apex of G(i+1) to a new vertex vi, 1 ≤ i ≤ n−1.
Theorem 4.5.4. The graph < K(1)1,n, K
(2)1,n > is a square DC graph.
Proof. Let G denote the graph < K(1)1,n, K
(2)1,n >.
Let V (G) = {y, v(1)i , v(2)i | 0 ≤ i ≤ n}, where v
(j)i are the pendant vertices of K
(j)1,n,
1 ≤ i ≤ n, j = 1, 2.
Let v(1)0 and v
(2)0 be the apex vertices of K
(1)1,n and K
(2)1,n respectively which are adjacent
to a new common vertex y.
Let E(G) = {v(1)0 v(1)i , v
(2)0 v
(2)i , yv
(1)0 , yv
(2)0 | 1 ≤ i ≤ n}.
It is to be noted that, |V (G)| = 2n+ 3 and |E(G)| = 2n+ 2.
Consider a bijection f : V (G)→ {1, 2, 3, . . . , 2n+ 3} is defined as below.
f(v(1)0 ) = 1.
f(v(2)0 ) = p,where p = max {x | x is a prime number and x ≤ 2n+ 3}.
56
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Allocate labels {2, 3, . . . , p − 1, p + 1, . . . , w} to the vertices v(1)1 , v
(1)2 , v
(1)3 , . . . v
(1)n ,
v(2)1 , v
(2)2 , v
(2)3 , . . . v
(2)n , y in any order, where w = 2n + 2 or 2n + 3 whichever is not
prime.
By looking into the above prescribed pattern,
ef (1) = n+ 1 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, < K(1)1,n, K
(2)1,n > is a square DC graph.
Example 4.5.4. Square DC labeling of the graph < K(1)1,5 , K
(2)1,5 > is demonstrated
in the following Figure 4.7.
2
1
4
3 6
5
7
8
10
9
12 11
13
Figure 4.7: Square DC labeling in < K(1)1,5 ,K
(2)1,5 >
Corollary 4.5.8. The graph < K(1)1,n, K
(2)1,n > is a CDC graph.
Corollary 4.5.9. The graph < K(1)1,n, K
(2)1,n > is a VODC graph.
Definition 4.5.5 ([13]). The graph constructed from given graph G by replacing
every edge ei of G by a graph K2,mifor some mi, 1 ≤ i ≤ q is called arbitrary
supersubdivision of G.
Theorem 4.5.5. Arbitrary supersubdivision of K1,n is a square DC graph.
Proof. V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and d(vj) = 1(1 ≤j ≤ n).
Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from arbitrary supersubdivision of
K1,n.
Then each edge v0vi of K1,n is exchanged by a graph K2,mifor some natural number
mi.
57
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Let uij be the vertices of mi vertex section, 1 ≤ i ≤ n, 1 ≤ j ≤ mi.
Let M =n∑i=1
mi.
It is to be noted that, |V (G)| = n+M + 1 and |E(G)| = 2M .
Consider a bijection f : V (G)→ {1, 2, 3, . . . , n+M + 1} is defined as below.
f(v0) = 1.
Label the vertices vi, 1 ≤ i ≤ n by the last n consecutive prime numbers between 1
to n + M + 1 respectively.
Allocate labels {2, 3, . . . , n + M + 1} to the vertices uij, 1 ≤ i ≤ n, 1 ≤ j ≤ mi in
any order.
By looking into the above prescribed pattern,
ef (1) = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, arbitrary supersubdivision of K1,n is a square DC graph.
Example 4.5.5. Square DC labeling of arbitrary supersubdivision of K1,4 is demon-
strated in the following Figure 4.8.
1
4 23
6
5
7
8
10
9
12
11
13
14
15
16
17
18
19
Figure 4.8: Square DC labeling in arbitrary supersubdivision of K1,4
58
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Remark 4.5.1. Arbitrary supersubdivision of K1,n is nothing but one point union
of the graphs K2,mi, where mi is arbitrary, 1 ≤ i ≤ n.
Corollary 4.5.10. Arbitrary supersubdivision of K1,n is a CDC graph.
Corollary 4.5.11. Arbitrary supersubdivision of K1,n is a VODC graph.
Theorem 4.5.6. The graph constructed from duplication of an edge in K1,n is a
square DC graph.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where d(v0) = n and d(vj) = 1(1 ≤ j ≤ n).
Let E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the resultant graph constructed from duplication of the edge e = v0vn
by a new edge e′ = v′0v′n, where
deg(vj) =
n; if j = 0.
1; if j = n.
2; if 1 ≤ j ≤ n− 1.
deg(v′j) =
n; if j = 0.
1; if j = n.
It is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n.
The labeling f : V (G)→ {1, 2, 3, . . . , n+ 3} is defined as below.
f(v0) = 1.
f(vn) = n+ 2.
f(v′n) = n+ 3.
f(v′0) = p, where p = max{x | x is a prime number and x ≤ n+ 3}.
Allocate labels {2, 3, . . . , p−1, p+1, . . . , n+1} to the vertices v1, v2, . . . , vn−1 in any
order.
By looking into the above prescribed pattern,
ef (1) = n = ef (0).
59
4.5. New Results on Square DC, Cube DC and Vertex Odd DC Labeling of Graphs
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed from duplication of an edge in K1,n is a square DC
graph.
Example 4.5.6. Square DC labeling of the graph constructed from duplication of
an edge in K1,8 is demonstrated in the following Figure 4.9.
1 9v0
v1
v2
v3
v4 v5
v6
v8
v7
v0 v8
v7
v1
v2
v3
v4
v5
v6
v'0v'8
2
1110
7
8
5
6
3
4
Figure 4.9: Square DC labeling in the graph constructed from duplication of an edge in K1,8
Corollary 4.5.12. The graph constructed from duplication of an edge in K1,n is a
CDC graph.
Corollary 4.5.13. The graph constructed from duplication of an edge in K1,n is a
VODC graph.
Definition 4.5.6 ([45]). Let V (Km,n) = {uj | 1 ≤ j ≤ m}⋃{vj | 1 ≤ j ≤ n}.The
graph Km,n�ui(K1) is defined by connecting a pendant vertex w to the vertex ui for
some i.
Theorem 4.5.7. K2,n � u2(K1) is a square DC graph.
Proof. Let G = K2,n � u2(K1).
Let V (G) = {u1, u2}⋃{vj, w | 1 ≤ j ≤ n}, where w is the pendant vertex adjacent
to u2 in G.
Let E(G) = {u1uj, u2uj, u2w | 1 ≤ j ≤ n}.Also it is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n+ 1.
60
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
The labeling f from V (G) to {1, 2, 3, . . . , n+ 3} is defined as below.
f(u1) = 1.
f(u2) = p,where p = max {x | x is a prime number and x ≤ n+ 3}.
Allocate labels {2, 3, . . . , p− 1, p+ 1, . . . , w} to the vertices w, vj(1 ≤ j ≤ n) in any
order, where w = n+ 2 or n+ 3 whichever is not prime.
By looking into the above prescribed pattern,
ef (1) = n, ef (0) = n+ 1.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K2,n � u2(K1) is a square DC graph.
Example 4.5.7. Square DC labeling in the graph K2,5� u2(K1) is demonstrated in
the following Figure 4.10.
1 7
8
42 3 65
u1 u2
v1 v2 v3 v4 v5
w
Figure 4.10: Square DC labeling in K2,5 � u2(K1)
Corollary 4.5.14. K2,n � u2(K1) is a CDC graph.
Corollary 4.5.15. K2,n � u2(K1) is a VODC graph.
4.6 VODC Labeling With the Use of Switching of a Vertex
in Cycle Allied Graphs
In this section, and in next two consecutive sections, VODC labeling of the graphs
constructed from the graph operation “switching of a vertex” is discussed. Particu-
61
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
larly in this section, we consider the switching of a vertex in cycle allied graphs.
Theorem 4.6.1. The graph Gv, where G is cycle Cn with one chord, is VODC.
Proof. Let G denote the graph cycle Cn with one chord.
Let V (G) = {vj | 1 ≤ j ≤ n} = V (Cn).
Let E(G) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}, where v2vn is the chord of
Cn. WLOG let the switched vertex be v1 (of degree 2 or degree 3).
Let Gv1 denote the graph constructed from switching of vertex v1 of G.
Corresponding to the vertices of different degree in Cn, it is required to discuss
following two cases.
Case 1: d(v1) = 2.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v1vj | 3 ≤ j ≤ n− 1}.
It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 4.
Consider a bijection f : V (Gv1)→ {1, 3, 5, . . . , 2n− 1} defined as below.
f(vj) =
2j − 1; 1 ≤ j ≤ 4.
2j + 1; 5 ≤ j ≤ n− 1.
f(vn) = 9.
If label of vn−1 is a multiple of 9 then interchange the labels of vn−1 and vn−2.
By looking into the above prescribed pattern,
ef (1) = n− 2 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(v1) = 3.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 2}.It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.
62
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Consider a bijection f : V (Gv1)→ {1, 3, 5, . . . , 2n− 1} defined as below.
f(vj) =
2j − 1; j = 1, 2, 6 ≤ j ≤ n.
2j − 3; 4 ≤ j ≤ 5.
f(v3) = 9.
By looking into the above prescribed pattern,
ef (1) = n− 3 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
That is, the graph Gv, where G is cycle Cn with one chord, is VODC.
Example 4.6.1. VODC labeling in (G)v, where d(v) = 2 and 3, G = C7 with one
chord, are demonstrated in the following Figure 4.11.
1
5
1
7
3
5v6
v5v4
v3
v2
v1
3
11
9 9
11
1313
7
v7
Figure 4.11: VODC labeling in (G)v.
Theorem 4.6.2. Gv is VODC, where G is cycle with twin chords Cn,3.
Proof. Let V (Cn,3) = {vj | 1 ≤ j ≤ n} = V (Cn).
Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}
⋃{v2vn−1}, where v2vn,
v2vn−1 are chords.
WLOG let v1 be the switched vertex.
Let (Cn,3)v1 denote the graph constructed from switching of vertex v1 of Cn,3.
Corresponding to the vertices of different degree in Cn,3, it is required to discuss
following three cases.
Case 1: d(v1) = 2.
63
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v2vn−1}
⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 3.
We define labeling function f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} as below.
f(vj) =
2j − 1; 1 ≤ j ≤ 4.
2j + 1; 5 ≤ j ≤ n− 1.
f(vn) = 9.
If label of vn−1 is a multiple of 9 then interchange the labels of vn−1 and vn−2.
By looking into the above prescribed pattern,
ef (1) = n− 2, ef (0) = n− 1.
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(v1) = 3.
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn}⋃{v1vj | 4 ≤ j ≤ n− 1}.
It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 5.
Consider a bijection f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} defined as below.
f(vj) =
2j − 1; j = 1, 2, 6 ≤ j ≤ n.
2j − 3; 4 ≤ j ≤ 5.
f(v3) = 9.
If label of vn−1 is a multiple of 3 then interchange the labels of vn−1 and vn−2.
By looking into the above prescribed pattern,
ef (1) =⌊2n−5
2
⌋, ef (0) =
⌈2n−5
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 3: d(v1) = 4.
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 3}.
64
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
It is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 7.
Consider a bijection f : V ((Cn,3)v1)→ {1, 3, 5, . . . , 2n− 1} defined as below.
f(vj) =
2j − 1; j = 1, 2, 6 ≤ j ≤ n.
2j − 3; 4 ≤ j ≤ 5.
f(v3) = 9.
By looking into the above prescribed pattern,
ef (1) =⌊2n−7
2
⌋, ef (0) =
⌈2n−7
2
⌉.
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Hence, the graph under consideration admits VODC labeling in each case.
That is, (Cn,3)v is VODC graph.
Example 4.6.2. C8,3 graph and VODC labeling in (C8,3)v, where d(v) = 2, 3, 4 are
demonstrated in the following Figure 4.12.
v4 v6
11
13
15 11
13
151
5
7
31
5
7
3
v3
v5
v7
v1
v2 1
3
7
511
9
9
9
13
15
v8
Figure 4.12: VODC labeling in (C8,3)v
Theorem 4.6.3. (Cn(1, 1, n− 5))v is VODC.
Proof. Let V (Cn(1, 1, n− 5)) = {vj | 1 ≤ j ≤ n} = V (Cn)
Let E(Cn(1, 1, n − 5)) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{v1v3}⋃{v3vn−1}
⋃{vn−1v1},where v1vn−1, v1v3, vn−1v3 are chords of Cn.
WLOG let v1 be the switched vertex.
Let (Cn(1, 1, n− 5))v1 denote the graph constructed from switching of a vertex v1 of
Cn(1, 1, n− 5).
Corresponding to the vertices of different degree in Cn(1, 1, n− 5), it is required to
discuss following two cases.
65
4.6. VODC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Case 1: d(v1) = 2.
Then by the effect of switching operation, the edge set of (Cn(1, 1, n− 5))v1 is
E((Cn(1, 1, n − 5))v1) = {vjvj+1 | 2 ≤ j ≤ n − 1}⋃{v2v4}⋃{v4vn−1}
⋃{vn−1v2}⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Cn(1, 1, n−5))v1)| = n and |E((Cn(1, 1, n−5))v1)| = 2n−2.
Let us a function f from V ((Cn(1, 1, n−5))v1) to {1, 3, 5, . . . , 2n−1} defined as below.
f(vj) =
2j − 3; j = 2, 3, 7 ≤ j ≤ 8.
2j − 5; 5 ≤ j ≤ 6.
2j − 1; 9 ≤ j ≤ n− 2.
f(v4) = 9.
f(vn−1) = 15.
f(vn) = p2;
f(v1) = p1;
where p1 = max{x | x is a prime number and x ≤ 2n− 1} and
p2 = max{y | y is a prime number and y < p1}.By looking into the above prescribed pattern,
ef (1) = n− 1 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(v1) = 4.
Then by the effect of switching operation, the edge set of (Cn(1, 1, n− 5))v1 is
E((Cn(1, 1, n−5))v1) = {vjvj+1 | 2 ≤ j ≤ n−1}⋃{v3vn−1}⋃{v1vj | 4 ≤ j ≤ n−2}.
It is to be noted that, |V ((Cn(1, 1, n−5))v1)| = n and |E((Cn(1, 1, n−5))v1)| = 2n−6.
66
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Consider a bijection f : V ((Cn(1, 1, n−5))v1)→ {1, 3, 5, . . . , 2n−1} defined as below.
f(vj) =
2j − 1; j = 1, 2, 7 ≤ j ≤ 8.
2j − 3; 4 ≤ j ≤ 6.
2j + 1; 9 ≤ j ≤ n− 2.
f(v3) = 9.
f(vn−1) = 15.
f(vn) = p, where p = max {x | x is a prime number and x ≤ 2n− 1}.
By looking into the above prescribed pattern,
ef (1) = n− 3 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
That is, (Cn(1, 1, n− 5))v is VODC.
Example 4.6.3. C8(1, 1, 3) graph and VODC labeling in (C8(1, 1, 3))v, where d(v) =
2, 4 are demonstrated in the following Figure 4.13.
v3
v5
v7
v8v2
v1 1
3
9
5
7
11
15
13 13
9
57
13
11
15
v4 v6
Figure 4.13: VODC labeling in (C8(1, 1, 3))v
4.7 VODC Labeling With the Use of Switching of a Vertex
in Wheel and Shell Allied Graphs
Theorem 4.7.1. (Wn)v is VODC, where v is rim vertex.
67
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Proof. Let V (Wn) = {vj | 0 ≤ j ≤ n}, where v0 is the apex vertex and vj(1 ≤ j ≤ n)
are the rim vertices of wheel Wn.
Let E(Wn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}.
Let (Wn)v1 denote the graph constructed from switching of a rim vertex v1 of Wn.
Then by the effect of switching operation, the edge set of (Wn)v1 is
E((Wn)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v0vj | 2 ≤ j ≤ n}⋃{v1vj | 3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Wn)v1)| = n+ 1 and |E((Wn)v1)| = 3n− 6.
Let us a function f : V ((Wn)v1))→ {1, 3, 5, . . . , 2n+ 1} defined as below.
Case 1: n ≤ 9.
f(v0) = 1.
f(vj) = 2j + 1; 1 ≤ j ≤ n.
Case 2: n > 9.
f(v0) = 1.
f(v1) = 3.
f(v2j) = pj+2; 1 ≤ j ≤ k,
f(v2j+1) = 3f(v2j); 1 ≤ j ≤ k,
where pj+2 = (j + 2)th prime number and k =⌊n−66
⌋.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−62
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌈3n−6
2
⌉, ef (0) =
⌊3n−6
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, (Wn)v is VODC, where v is rim vertex.
Example 4.7.1. Wheel graph W9 and VODC labeling in (W9)v1, where v1 is rim
vertex are demonstrated in the following Figure 4.14.
68
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
v1
v2
v3
v4
v5 v6
v7
v8
v9
v0 1
53
9
7
11
19
17
15
13
Figure 4.14: VODC labeling in (W9)v1
Theorem 4.7.2. Gv is VODC, where G is gear graph Gn and v is not apex vertex.
Proof. Let V (Gn) = {v0}⋃{vj | 1 ≤ j ≤ 2n}, where d(v0) = n and vj(1 ≤ j ≤ 2n)
are other vertices of gear graph Gn,
deg(vj) =
2 when j is even;
3 when j is odd; 1 ≤ j ≤ 2n.
Let E(Gn) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 1 ≤ j ≤ 2n− 1}⋃{u2nu1}.(Gn)ui
∼= (Gn)uj , where d(ui) = d(uj).
Let (Gn)ui denote the graph constructed from switching of vertex uj (j = 1, 2) of
Gn.
Corresponding to the vertices of different degree in Gn, it is required to discuss
following two cases.
Case 1: deg(u1) = 3.
Then by the effect of switching operation, the edge set of (Gn)u1 is
E((Gn)u1) = {u0u2j−1 | 2 ≤ j ≤ n}⋃{ujuj+1 | 2 ≤ j ≤ 2n − 1}⋃{u1uj | 3 ≤ j ≤2n− 1}.It is to be noted that, |V (Gn)u1| = 2n+ 1 and |E(Gn)u1| = 5n− 6.
Consider a bijection f : V ((Gn)u1)→ {1, 3, . . . , 4n+ 1} defined as below.
Let f(u1) = 1
69
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
which will generate 2n− 3 edges (which are adjacent with v1) with label 1.
Let f(u0) = 3.
Now it remains to generate k = 5n−62− (2n− 3) edges with label 1.
For the vertices u3, u5, . . . , uk, allocate the vertex labels as per following ordered
pattern upto it generate k edges with label 1.
f(u2j+1) = 3(2j + 1); 1 ≤ j ≤ k,
where k =
n2; if n is even.
n+12
; if n is odd.
Allocate labels {5, 7, 11, . . . , 2n} to the vertices {u2, u4, . . .} and {uk+1, uk+2, . . . , u2n}in any order.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n is even ef (1) = 5n−62
= ef (0)
n is odd ef (1) = 5n−52, ef (0) = 5n−7
2
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: deg(u2) = 2.
Then by the effect of switching operation, the edge set of (Gn)u2 is
E((Gn)u2) = {u0u2j−1 | 1 ≤ j ≤ n}⋃{ujuj+1 | 3 ≤ j ≤ 2n − 1}⋃{u2nu1}⋃{u2uj |
4 ≤ j ≤ 2n}.It is to be noted that, |V (Gn))u2 | = 2n+ 1 and |E(Gn))u2| = 5n− 4.
Consider a bijection f : V ((Gn)u2)→ {1, 3, . . . , 4n+ 1} defined as below.
Let f(u1) = 1
which will generate 2n− 3 edges (which are adjacent with v1) with label 1.
Let f(u0) = 3.
Now it remains to generate k = 5n−62− (2n− 3) edges with label 1.
70
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
For the vertices u2, u4, . . . , uk, allocate the vertex labels as per following ordered
pattern upto it generate k edges with label 1.
f(u2j) = 3(2j + 1); 1 ≤ j ≤ k,
where k =
n2; if n is even.
n+12
; if n is odd.
Allocate labels {5, 7, 11, . . . , 2n} to the vertices {u3, u5, . . .} and {uk+1, uk+2, . . . , u2n}in any order.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n is even ef (1) = 5n−42
= ef (0)
n is odd ef (1) = 5n−32, ef (0) = 5n−5
2
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Hence, the graph under consideration admits VODC labeling.
That is, (Gn)v (v is not apex vertex) is VODC.
Example 4.7.2. VODC labeling in (G6)v, where d(v) = 3, d(v) = 2 are demon-
strated in the following Figure 4.15.
3
13
7
1
59
3
11
7
913
1
15 17
17
19
19
21
21
15
23
2523
5
11
u12 25
u0
u2
u3
u4
u5
u6
u7
u8
u9
u10
u1
u11
Figure 4.15: VODC labeling in (G6)v
Theorem 4.7.3. Gv is VODC, where G is shell graph Sn and v is not apex vertex.
Proof. Let V (Sn) = {uj | 0 ≤ j ≤ n − 1}, where u0 is the apex vertex and uj(1 ≤j ≤ n− 1) are the other vertices of shell Sn, where
71
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
deg(uj) =
2 when j = 1, n− 1.
3 when 2 ≤ j ≤ n− 2.
Let E(Sn) = {u0uj | 2 ≤ j ≤ n− 2}⋃{ujuj+1 | 0 ≤ j ≤ n− 2}⋃{u0un−1}.(Sn)ui
∼= (Sn)uj , where d(ui) = d(uj).
Let (Sn)ui denote the graph constructed from switching of vertex uj (j = 1, 2) of
Sn.
Corresponding to the vertices of different degree in Sn, it is required to discuss fol-
lowing two cases.
Case 1: d(u2) = 3.
Then by the effect of switching operation, the edge set of (Sn)u2 is
E((Sn)u2) = {u0uj | 3 ≤ j ≤ n−2}⋃{ujuj+1 | 3 ≤ j ≤ n−1}⋃{un−1u0}{u0u1}⋃{u2uj |
4 ≤ j ≤ n− 1}.It is to be noted that, |V ((Sn)u2)| = n and |E((Sn)u2)| = 3n− 10.
Consider a bijection f : V ((Sn)u2)→ {1, 3, . . . 2n− 1} as per the following subcases.
Subcase 1: n ≤ 15.
f(u0) = 1.
f(u1) = p,where p = max {x | x is a prime number and x ≤ 2n− 1}.
f(u2) = 3.
f(uj) = 2j − 1; 3 ≤ j ≤ n− 1.
Subcase 2: n > 15.
f(u0) = 1.
f(u1) = p,where p = max {x | x is a prime number and x ≤ 2n− 1}.
f(u2) = 3.
f(u2j−1) = pj; 2 ≤ j ≤ k,
f(u2j) = 3f(u2j−1); 2 ≤ j ≤ k,
where pj denotes the jth prime number and k =⌊n−10
6
⌋.
By looking into the above prescribed pattern,
72
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−102
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 3n−92, ef (0) = 3n−11
2
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(u1) = 2.
Then by the effect of switching operation, the edge set of (Sn)u1 is
E((Sn)u1) = {u0uj | 2 ≤ j ≤ n − 1}⋃{ujuj+1 | 2 ≤ j ≤ n − 1}⋃{un−1u0}{u1uj |3 ≤ j ≤ n− 1}.It is to be noted that, |V ((Sn)u1)| = n, |E((Sn)u1)| = 3n− 8.
Consider a bijection f : V ((Sn)u1)→ {1, 3, . . . 2n− 1} defined as below.
Subcase 1: n ≤ 9.
f(uj) = 2j + 1; 0 ≤ j ≤ n− 1.
Subcase 2: n > 9.
f(u0) = 1.
f(u1) = 3.
f(u2j) = pj+2; 1 ≤ j ≤ k,
f(u2j+1) = 3f(u2j); 1 ≤ j ≤ k,
where pj+2 is (j + 2)th prime number and k =⌊n−46
⌋.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−82
= ef (0)
n ≡ 1, 3(mod 4) ef (1) = 3n−72, ef (0) = 3n−9
2
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
That is, (Sn)v is VODC and v is not apex vertex.
Example 4.7.3. Shell graph S7 and VODC labeling in (S7)v, where d(v) = 3, d(v) =
2 are demonstrated in the following Figure 4.16.
73
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
1
3
5
u2
u3 u4
u5
u1
u0
u6
1
3
5
7
9
9
13
7
13 11
11
Figure 4.16: VODC labeling in (S7)v
Theorem 4.7.4. (Fln)v is VODC, where v is not apex vertex.
Proof. Let V (Fln) = {u0}⋃{uj | 1 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where u0 is the
apex vertex, uj(1 ≤ j ≤ n) are the internal vertices and vj(1 ≤ j ≤ n) are the
external vertices; deg(ui) = 4 and deg(vi) = 2.
Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤
j ≤ n}⋃{ujvj | 1 ≤ j ≤ n}, where ujvj are the spoke edges.
(Fln)ui∼= (Fln)uj , where d(ui) = d(uj).
Let (Fln)u1 and (Fln)v1 denote the graph constructed from switching of vertex u1
and v1 of Fln respectively.
Corresponding to the vertices of different degree in Fln, it is required to discuss
following two cases.
Case 1: d(u1) = 4.
By the effect of switching operation, the edge set of (Fln)u1 is
E((Fln)u1) = {ujuj+1 | 2 ≤ j ≤ n}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.It is to be noted that, |V ((Fln)u1)| = 2n+ 1, |E((Fln)u1)| = 6n− 8.
Consider a bijection f : V ((Fln)u1)→ {1, 3, . . . 4n+ 1} defined as below.
Subcase 1: n ≤ 8.
f(u0) = 1.
f(u1) = 3.
f(v1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.
74
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
For the vertices u2, u3, . . . , un, v2, v3, . . . , vn of (Fln)u1 allocate the vertex labels such
that for any edge ujuj+1 ∈ E(Fln)u1 ,
f(uj) - f(uj+1)
f(uj) - f(vj); 2 ≤ j ≤ n.
Subcase 2: n > 8.
f(u0) = 1.
f(u1) = 3.
f(v1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.
f(uj) = pj; 2 ≤ j ≤ k,
f(vj) = 3f(uj); 2 ≤ j ≤ k,
where pj is jth prime number and k =⌊n3
⌋− 2.
For the vertices uk+1, uk+2, . . . , un, vk+1, vk+2, . . . , vn of (Fln)u1 , allocate the vertex
labels such that for any edge ujuj+1 ∈ E(Fln)u1 ,
f(uj) - f(uj+1); k + 1 ≤ j ≤ n− 1
f(uj) - f(vj); k + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 4 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
Case 2: d(v1) = 2.
Then by the effect of switching operation, the edge set of (Fln)v1 is
E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{v1uj | 2 ≤ j ≤ n− 1}⋃{v1vj | 2 ≤ j ≤ n}.It is to be noted that, |V ((Fln)v1)| = 2n+ 1, |E((Fln)v1)| = 6n− 4.
75
4.7. VODC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Consider a bijection f : V ((Fln)v1)→ {1, 3, . . . 4n+ 1} defined as below.
f(u0) = 1.
f(v1) = 3.
f(u1) = p,where p = max {x | x is a prime number and x ≤ 4n+ 1}.
f(uj) = pj+1; 2 ≤ j ≤ k,
f(vj) = 3f(uj); 2 ≤ j ≤ k,
where pj+1 = (j + 1)th prime number and k =⌊n3
⌋.
Allocate the labels to vertices uk+1, uk+2, . . . , un, vk+1, vk+2, . . . , vn of (Fln)v1 such
that
f(uj) - f(uj+1); k + 1 ≤ j ≤ n− 1
f(uj) - f(vj); k + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 2 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in this case.
That is, (Fln)v is VODC (v is not apex vertex) .
Example 4.7.4. Flower graph Fl4 and VODC labeling in (Fl4)v, where d(v) = 2
and 4 are demonstrated in the following Figure 4.17.
3
9
13
11
75
17
15
1
u4
v2v3
u2 u3
u1
v1v4
u01
9
7
3
5
11 13
1715
Figure 4.17: VODC labeling in (Fl4)v
76
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
4.8 VODC Labeling With the Use of Ringsum of Different
Graphs with Star Graph K1,n
Ghodasara and Rokad[5] illumined and derived some captivating results on cordial
labeling of the graphs by considering ringsum of K1,n with different graph families.
Under the inspiration of this credibility, in current segment we demonstrate few
graphs constructed from the graph operation ringsum for VODC labeling.
Theorem 4.8.1. G⊕K1,n is VODC graph, where G is cycle Cn with one chord.
Proof. Let G denote the cycle Cn with one chord.
Let V (G ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u2un},
where u2un is the chord of Cn and the egdes u1u2, u2un, u1un form a triangle.
It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 1.
Consider a bijection f : V (G⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.
f(u1) = 3.
f(u2) = 1.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =
⌊4n− 3
6
⌋.
f(u2j+1) = pj+2; 1 ≤ j ≤ k2,
f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,
where pj+2 is (j + 2)th prime number and k2 =⌊n−53
⌋.
For the vertices u2k2+3, u2k2+4, . . . , un and vk1+1, vk1+2, . . . , vn allocate the vertex la-
bels such that for any edge ujuj+1 ∈ E(G⊕K1,n),
f(uj) - f(ui+1), 2k2 + 3 ≤ i ≤ n;
f(u1) - f(vi), k + 1 ≤ i ≤ n.
77
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
By looking into the above prescribed pattern,
ef (0) = n+ 1, ef (1) = n.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, G⊕K1,n is a VODC graph, where G is the cycle Cn with one chord.
Example 4.8.1. VODC labeling in ringsum of C7 with one chord and K1,7 is demon-
strated in Figure 4.18.
v3 v4 v5 v6 v7v1 v2
3
5
7
9
11
u3
u4 u5
u6
u7
u1
u2 117
15 1921 27 2523
13
Figure 4.18: VODC labeling in the graph constructed from ringsum of C7 with one chord and K1,7
Theorem 4.8.2. Cn,3 ⊕K1,n is a VODC graph.
Proof. Let V (Cn,3 ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn,3) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Cn,3 ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤
n}⋃{u2un}⋃{u2un−1}, where unu2 and u2un−1 are the chords.
It is to be noted that, |V (Cn,3 ⊕K1,n)| = 2n and |E(Cn,3 ⊕K1,n)| = 2n+ 2.
Consider a bijection f : V (Cn,3 ⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.
f(u1) = 3.
f(u2) = 1.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, k1 =
⌊4n− 3
6
⌋.
f(u2j+1) = pj+2; 1 ≤ i ≤ k2,
f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,
78
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
where pj+2 is (j + 2)th prime number and k2 =⌊n−53
⌋.
For the vertices u2k2+3, u2k2+4, . . . , un and vk1+1, vk1+2, . . . , vn allocate the vertex la-
bels such that for any edge e = ujuj+1 ∈ E(Cn,3 ⊕K1,n),
f(uj) - f(uj+1), 2k2 + 3 ≤ j ≤ n.
f(u1) - f(vj), k1 + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (0) = n+ 1 = ef (1).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn,3 ⊕K1,n is a VODC graph.
Example 4.8.2. VODC labeling in C8,3⊕K1,8 is demonstrated in the Figure 4.19.
v3 v6v1
u3
u6
u7
v8
u2 u81
5
7
9
11
13
15
3
v5 v7
u5
u4
u1
25
17
21 27 19 23 29` 31
v4v2
Figure 4.19: VODC labeling in C8,3 ⊕K1,8
Theorem 4.8.3. Cn(1, 1, n− 5)⊕K1,n is a VODC graph.
Proof. Let G denote cycle with triangle Cn(1, 1, n− 5).
Let V (G ⊕ K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = {uj | 1 ≤ j ≤ n} and
V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤
n}⋃{u1u3}⋃ {u3un−1}
⋃{un−1u1}, where u1, u3 and un−1 are the vertices of tri-
angle formed by the chords u1u3, u3un−1 and u1un−1.
79
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 3.
Consider a bijection f : V (G⊕K1,n)→ {1, 3, . . . , 4n− 1} defined as below.
f(u1) = 3.
f(u3) = 1.
f(u2) = p,where p = max {x | x is the largest prime number x ≤ 4n− 1},
f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =
⌊4n− 3
6
⌋.
f(u2(j+1)) = pj+2; 1 ≤ j ≤ k2,
f(u2j+3) = 5f(u2(j+1)); 1 ≤ j ≤ k2,
where pj+2 is (j + 2)th prime number and k2 =⌊n−53
⌋.
For the vertices u2k2+4, u2k2+5, . . . , un and vk1+1, vk1+2, . . . , vn allocate the vertex la-
bels such that for any edge ujuj+1 ∈ E(G⊕K1,n),
f(uj) - f(uj+1), 2k2 + 4 ≤ j ≤ n.
f(v) - f(vj), k1 + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (0) = n+ 1, ef (1) = n+ 2.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn(1, 1, n− 5)⊕K1,n is a VODC graph.
Example 4.8.3. VODC labeling in the graph C8(1, 1, 3) ⊕K1,8 is demonstrated in
the following Figure 4.20.
80
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
v3v1
u3
u4
u6
u7
u2
u1
u5
u8
v2 v4 v5 v6 v7 v8
13
159
11
5 7
1
3
25
21 27 17 19 23 29
31
Figure 4.20: VODC labeling in C8(1, 1, 3)⊕K1,8
Theorem 4.8.4. Pn ⊕K1,n is a VODC graph.
Proof. Let V (Pn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Pn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Pn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Pn ⊕K1,n)| = 2n and |E(Pn ⊕K1,n)| = 2n− 1.
Consider a bijection f : V (Pn ⊕K1,n)→ {1, 3, 5, . . . , 4n− 1} defined as below.
f(u1) = 3.
f(u2) = 1.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k1, where k1 =
⌊4n− 3
6
⌋.
f(u2j+1) = pj+2; 1 ≤ j ≤ k2,
f(u2j+2) = 5f(u2j+1); 1 ≤ j ≤ k2,
where pj+2 is (j + 2)th prime number and k2 =⌊n−53
⌋.
For the vertices u2k2+3, u2k2+4, . . . , un and vk1+1, vk1+2, . . . , vn allocate the vertex la-
bels such that for any edge e = ujuj+1 ∈ E(Pn ⊕K1,n),
f(ui) - f(ui+1), 2k2 + 3 ≤ i ≤ n.
f(v) - f(vj), k1 + 1 ≤ j ≤ n.
81
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
By looking into the above prescribed pattern,
ef (1) = n− 1, ef (0) = n.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Pn ⊕K1,n is a VODC graph.
Example 4.8.4. VODC labeling in the graph P5 ⊕ K1,5 is demonstrated in the
following Figure 4.21.
u1u2u3u4u5
v2
v3
v4
57
v1
v5
31
9
11
15
13
1719
Figure 4.21: VODC labeling in P5 ⊕K1,5
Theorem 4.8.5. Wn ⊕K1,n is a VODC graph.
Proof. Let V (Wn ⊕K1,n) = {u, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u, uj | 1 ≤ j ≤n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here u is the apex vertex, ui(1 ≤ i ≤ n) are rim vertices of Wn and vi(1 ≤ i ≤ n)
are the pendant vertices, u1 is apex vertex of star graph.
Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{uuj | 1 ≤ j ≤ n}⋃{u1vj |
1 ≤ j ≤ n}.It is to be noted that, |V (Wn ⊕K1,n)| = 2n+ 1 and |E(Wn ⊕K1,n)| = 3n.
Consider a bijection f : V (Wn ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 1} defined as below.
f(u1) = 3.
f(u) = 1.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k,
f(uj+1) = pj+2; 1 ≤ j ≤ k,
where pj+2 is (j + 2)th prime number and k =⌊n2
⌋.
For the vertices uk+2, uk+3, . . . , un, vk+1, vk+2, . . . , vn allocate the vertex labels such
82
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
that for any edge ujuj+1 ∈ E(Wn ⊕K1,n),
f(uj) - f(uj+1), k + 2 ≤ j ≤ n.
f(u1) - f(vj), k + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌈3n
2
⌉, ef (0) =
⌊3n
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Wn ⊕K1,n is a VODC graph.
Example 4.8.5. VODC labeling in the graph W6 ⊕ K1,6 is demonstrated in the
following Figure 4.22.
u
v6
u3
u4
u5
u1
u2
1
3
5
7
9
11
13
17
15 21 19 2523
v1 v2 v4v3 v5
u6
Figure 4.22: VODC labeling in W6 ⊕K1,6
Theorem 4.8.6. Fln ⊕K1,n is a VODC graph.
Proof. Let V (Fln ⊕K1,n) = {u, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are
external vertices of Fln; d(vj) = 1(1 ≤ j ≤ n), w1 is apex vertex of star graph.
Let E(Fln⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{uuj | 1 ≤ j ≤ n}⋃{uwj |
1 ≤ j ≤ n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.
83
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
Consider a bijection f : V (Fln ⊕K1,n)→ {1, 3, 5, . . . , 6n+ 1} defined as below.
f(u) = 1.
f(w1) = 3.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k, where k =⌈n
2
⌉.
For the vertices vk+1, vk+2, . . . , vn and u1, u2 . . . un, w2, w3 . . . wn allocate the vertex
labels such that for any edge ujuj+1 ∈ E(Fln ⊕K1,n),
f(w1) - f(vj); k + 1 ≤ j ≤ n,
f(uj) - f(uj+1), f(uj) - f(wj); 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌈5n
2
⌉, ef (0) =
⌊5n
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Fln ⊕K1,n is a VODC.
Example 4.8.6. VODC labeling in the graph Fl4 ⊕ K1,4 is demonstrated in the
following Figure 4.23.
u1
u2
u3
u4
w1
v1
v2 v3
v4
w4
w2
w3
u1
3
79
5
1113
15
17
19
21 23
25
Figure 4.23: VODC labeling in Fl4 ⊕K1,4
Theorem 4.8.7. K2,n ⊕K1,n is a VODC graph.
84
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
Proof. Let V (K2,n ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (K2,n) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.
Consider a bijection f : V (K2,n ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 3} defined as below.
f(u) = 1.
f(w) = p.
f(u1) = 3.
f(vj) = 3(2j + 1); 1 ≤ j ≤ k,
f(uj+1) = pj+2; 1 ≤ j ≤ n− 1,
where, p = max {x | x is the largest prime number x ≤ 4n+ 3},pj+2 is (j + 2)th prime number and k =
⌈n2
⌉.
For the vertices vk+1, vk+2, . . . , vn allocate the vertex labels s. t.
f(v) - f(vj), k + 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = 3n2
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌈3n2
⌉, ef (0) =
⌊3n2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K2,n ⊕K1,n is a VODC graph.
Example 4.8.7. VODC labeling in the graph K2,7 ⊕ K1,7 is demonstrated in the
following Figure 4.24.
85
4.8. VODC Labeling With the Use of Ringsum of Different Graphs with Star Graph K1,n
17
19
29
u1
u2
v1
v2
u3
v3 v4
v5
u4
w
u
u5
u6
u7
v6
v7
1
31
5
7
11
13
9
1521 27
25
23
3
Figure 4.24: VODC labeling in K2,7 ⊕K1,7
Theorem 4.8.8. DFn ⊕K1,n is a VODC graph.
Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u, vj | 1 ≤ j ≤ n}.Here u,w are apex vertices of DFn, u1, u2, . . . , un are vertices of path Pn correspond-
ing to DFn, d(vj) = 1, where 1 ≤ j ≤ n and u is apex vertex of star graph.
Let E(DFn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uvj | 1 ≤ j ≤ n}⋃{uuj | 1 ≤ j ≤n}⋃{wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn ⊕K1,n)| = 2n+ 2 and |E(DFn ⊕K1,n)| = 4n− 1.
Consider a bijection f : V (DFn ⊕K1,n)→ {1, 3, 5, . . . , 4n+ 3} defined as below.
f(w) = 3.
f(u) = 1.
f(uj) = pj+2; 1 ≤ j ≤ n,
f(vj) = 3(2j + 1); 1 ≤ j ≤ k,
where pj+2 is (j + 2)th prime number and k =⌊n2
⌋.
For the vertices vk+1, vk+2, . . . , vn allocate the vertex labels such that
f(uj) - f(uj+1), 2k + 1 ≤ j ≤ n.
86
4.9. Conclusion and Scope for Further Research
By looking into the above prescribed pattern,
ef (1) = 2n, ef (0) = 2n− 1.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DFn ⊕K1,n is a VODC graph.
Example 4.8.8. VODC labeling in the graph DF5 ⊕ K1,5 is demonstrated in the
following Figure 4.25.
u1 u2 u3 u4 u5
9v1
v2
v3
v4
v5
u
w
5 11 13 17
19
23
3
1
15
7
21
Figure 4.25: VODC labeling in DF5 ⊕K1,5
4.9 Conclusion and Scope for Further Research
In this chapter we have emanated some new square DC, CDC and VODC graphs. We
have derived new results on these labelings using graph operations such that one
point union, arbitrary supersubdivision and duplication. We have also emanated
VODC labeling for larger graphs which are constructed from a standard graph by
means of graph operations such as ringsum and switching of a vertex.
However, there is no such standard relation between either of the three Square DC,
CDC and VODC labelings; like, we may find a graph which admits one of the three
labelings but not the other. To observe this matter more effectively, the list of graph
families satisfying/ not satisfying certain labeling is shown in following table.
87
4.9. Conclusion and Scope for Further Research
Graph Square DC CDC VODC
Wn Yes Yes Yes
P7 Yes No No
K4 No Yes No
Wn ⊕K1,n No No Yes
K1,n(n ≥ 8) No No No
Still there is a big scope of research and extension/generalization of the results de-
rived here. Here we propose some open problems which may provide better direction
for further delevopment in these labelings.
Problem 4.9.1. Derive essential and adequate condition (if any) for any graph to
be a VODC graph.
Problem 4.9.2. Investigate some new square DC, CDC, VODC graphs with respect
to other graph operations.
Problem 4.9.3. Classify/Generalize the graph G such that G ⊕ K1,n is a square
DC, CDC, VODC graph. (Here |V (G)| may or may not be equal to n.)
In the next chapter, we will discuss sum DC labeling of graphs.
88
CHAPTER 5
Sum Divisor Cordial Labeling of
Graphs
Different variants of DC labeling namely square DC, CDC and VODC labeling were
discussed in the earlier Chapter-4, while the current chapter aims to deliberate
another labeling having DC theme.
Inspired by the idea of DC labeling in 2016, A. Lourdusamy and F. Patrick originated
the concept of one of the variant of DC labeling called sum DC labeling.
5.1 Introduction
Definition 5.1.1 (Lourdusamy et al.[1]). Let G = (V,E) be a simple graph with
order p and size q. Consider a bijection f : V (G) → {1, 2, . . . , |V (G)|} and let the
edge labeling function f ∗ : E(G)→ {0, 1} be defined as
f ∗(e = uv) =
1; if 2 | [f(u) + f(v)]
0; otherwise
Then the function f is called a sum DC labeling if |ef (0)− ef (1)| ≤ 1.
A graph which confesses SDC labeling is called SDC graph.
Example 5.1.1. A SDC labeling of star graph K1,7 is demonstrated in the following
Figure 5.1.
89
5.2. Some Existing Results on Sum DC Labeling
1
2
3
4 6
5
8
7v0
v1
v2
v3
v4
v5
v6
v7
Figure 5.1: SDC labeling in K1,7
It is clear that for a graph, we may find more than one SDC labeling by using proper
permutation of the labels used (such that the condition for edge labels is satisfied).
5.2 Some Existing Results on Sum DC Labeling
SDC labeling was introduced in 2016. Since then, many researchers have explored
this labeling by finding captivating results during last three years.
In first paper on SDC labeling[1], Lourdusamy et al. have established SDC labeling
for some basic graphs and derived the following results.
Theorem 5.2.1. The path Pn is SDC.
Theorem 5.2.2. The comb Pn �K1 and the crown Cn �K1 are SDC graph.
Theorem 5.2.3. The star graph K1,n and the barycentric subdivision of the star
K1,n are SDC.
Theorem 5.2.4. The complete bipartite graph K2,n is SDC.
Theorem 5.2.5. The graph K2 +mK1 is SDC graph.
Theorem 5.2.6. The bistar Bn,n and restricted square of bistar Bn,n are SDC.
Theorem 5.2.7. The flower graph Fln, gear graph Gn and the jewel graph Jn are
SDC.
In another research paper[2], the same authors have derived more results in which
characterize a SDC graph. They have also developed SDC labeling in some K1,n
and Bm,n allied graphs. These results are listed below.
90
5.3. Some New Cycle Related Sum DC Graphs
Theorem 5.2.8. The barycentric subdivision of bistar Bn,n is SDC.
Theorem 5.2.9. S ′(K1,n) and D2(K1,n) of star K1,n are SDC.
Theorem 5.2.10. S ′(Bn,n) and DS(Bn,n) of bistar Bn,n are SDC.
Theorem 5.2.11. D2(Bn,n) is SDC.
Theorem 5.2.12. The closed helm graph CHn is SDC graph.
Rozario and Surya[38] have discussed SDC labeling for switching invariance in path
Pn and cycle Cn. These results are listed below.
Theorem 5.2.13. For n ∈ N, there is a SDC graph G which has n vertices.
Theorem 5.2.14. Switching invarivance in path Pn is SDC.
Theorem 5.2.15. Switching invarivance in cycle Cn is SDC.
5.3 Some New Cycle Related Sum DC Graphs
A. Lourdusamy and F. Patrick[1] demonstrate some fascinating graphs for SDC
labeling. Under the motivation of this belief, in current section we originate SDC
labeling of some cycle related graphs.
Theorem 5.3.1. Cycle Cn is SDC graph for n ≡ 0, 1, 3(mod 4).
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}. It is to be noted that, |V (Cn)| = n = |E(Cn)|.Consider a bijection f : V (Cn)→ {1, 2, 3, . . . , n} defined as below.
f(vj) =
j; j ≡ 0, 1(mod 4).
j + 1; j ≡ 2(mod 4).
j − 1; j ≡ 3(mod 4), 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0(mod 4) ef (0) = ef (1) = n2
n ≡ 1(mod 4) ef (0) =⌊n2
⌋, ef (1) =
⌈n2
⌉
n ≡ 3(mod 4) ef (0) =⌈n2
⌉, ef (1) =
⌊n2
⌋
91
5.3. Some New Cycle Related Sum DC Graphs
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn is SDC graph.
Example 5.3.1. SDC labeling in cycle C5 is demonstrated in the following Figure
5.2.
1
3
2 4
5
v1
v2
v3v4
v5
Figure 5.2: SDC labeling in C5
Theorem 5.3.2. Cn with one chord is SDC graph.
Proof. Let cycle Cn with one chord be denoted as G.
Let V (G) = {vj | 1 ≤ j ≤ n} and E(G) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}⋃{v2vn},
where v2vn is a chord.
Also it is to be noted that, |V (G)| = n and |E(G)| = n+ 1.
Consider a bijection f : V (G)→ {1, 2, 3, . . . , n} defined as below.
Case 1: n ≡ 0, 1, 2(mod 4).
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.
Case 2: n ≡ 3(mod 4).
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 2.
f(vn−1) = n.
f(vn) = n− 1.
By looking into the above prescribed pattern,
92
5.3. Some New Cycle Related Sum DC Graphs
Cases of n Edge conditions
n ≡ 0, 2, 3(mod 4) ef (1) =⌊n+12
⌋, ef (0) =
⌈n+12
⌉
n ≡ 1(mod 4) ef (0) = ef (1) = n+12
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn with one chord is SDC graph.
Example 5.3.2. SDC labeling in C6 with one chord is demonstrated in the following
Figure 5.3.
v1
1
2
4
3
5
6
v3
v4
v5
v6v2
Figure 5.3: SDC labeling in C6 with one chord
Theorem 5.3.3. Cycle with twin chords Cn,3 is SDC graph.
Proof. Let V (Cn,3) = {vj | 1 ≤ j ≤ n}.Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}
⋃{v2vn}⋃{v2vn−1}, where vnv2,
v2vn−1 are the chords.
It is to be noted that, |V (Cn,3)| = n and |E(Cn,3)| = n+ 2.
Consider a bijection f : V (Cn,3)→ {1, 2, 3, . . . , n} defined as below.
Case 1: n ≡ 0(mod 4).
f(vj) =
j; j ≡ 1, 0(mod 4).
j + 1; j ≡ 2(mod 4).
j − 1; j ≡ 0(mod 4) 1 ≤ j ≤ n.
Case 2: n ≡ 1, 2(mod 4).
f(vi) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.
93
5.3. Some New Cycle Related Sum DC Graphs
Case 3: n ≡ 3(mod 4).
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 2.
f(vn−1) = n.
f(vn) = n− 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) = n+22
= ef (0)
n ≡ 1(mod 4) ef (0) =⌊n+22
⌋, ef (1) =
⌈n+22
⌉
n ≡ 3(mod 4) ef (1) =⌊n+22
⌋, ef (0) =
⌈n+22
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn,3 is SDC graph.
Example 5.3.3. SDC labeling in C7,3 is demonstrated in the following Figure 5.4.
1
2
4
3 5
6
7
v7
v6
v5v4
v3
v2
v1
Figure 5.4: SDC labeling in C7,3
Theorem 5.3.4. Cn(1, 1, n− 5) is SDC graph for n ≡ 0, 1, 2(mod 4).
Proof. Let V (Cn(1, 1, n− 5)) = {vj | 1 ≤ j ≤ n}.Let E(Cn(1, 1, n−5)) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}
⋃{v1v3}⋃{v3vn−1}
⋃{vn−1v1},where v1vn−1, v1v3, vn−1v3 are the chords of Cn(1, 1, n− 5).
It is to be noted that, |V (Cn(1, 1, n− 5))| = n and |E(Cn)| = n+ 3.
Consider a bijection f : V (Cn(1, 1, n− 5))→ {1, 2, 3, . . . , n} defined as below.
94
5.3. Some New Cycle Related Sum DC Graphs
Case 1: n ≡ 0, 1(mod 4).
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(vj) =
j; j ≡ 1, 0(mod 4).
j + 1; j ≡ 2(mod 4).
j − 1; j ≡ 3(mod 4), 1 ≤ j ≤ n− 2.
f(vn−1) = n.
f(vn) = n− 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1(mod 4) ef (0) = n+32
= ef (1)
n ≡ 0(mod 4) ef (1) =⌊n+32
⌋, ef (0) =
⌈n+32
⌉
n ≡ 2(mod 4) ef (0) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn(1, 1, n− 5) is SDC graph.
Example 5.3.4. SDC labeling in C8(1, 1, 3) is demonstrated in the following Figure
5.5.
1
2
4
3
5
6
8
7
v3
v4
v5
v6
v7
v8v2
v1
Figure 5.5: SDC labeling in C8(1, 1, 3)
95
5.3. Some New Cycle Related Sum DC Graphs
Theorem 5.3.5. Wn is SDC graph for n ≡ 0, 1, 2(mod 4).
Proof. Let V (Wn) = {v0, vj, | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are rim vertices and
v0 is the apex vertex.
Let E(Wn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}.
It is to be noted that, |V (Wn)| = n+ 1 and |E(Wn)| = 2n.
Consider a bijection f : V (Wn)→ {1, 2, 3, . . . , n+ 1} defined as below.
Case 1: n ≡ 1(mod 4).
f(v0) = 1.
f(vj) =
j; j ≡ 3(mod 4).
j + 1; j ≡ 0, 1(mod 4).
j + 2; j ≡ 2(mod 4), 1 ≤ j ≤ n.
Case 2: n ≡ 0(mod 4).
f(v0) = 1.
f(vj) =
j; j ≡ 3(mod 4).
j + 1; j ≡ 0, 1(mod 4).
j + 2; j ≡ 2(mod 4), 1 ≤ j ≤ n.
Case 3: n ≡ 2(mod 4).
f(v0) = 2.
f(vj) =
j; j ≡ 1(mod 4).
j + 1; j ≡ 2, 3(mod 4).
j + 2; j ≡ 0(mod 4), 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 1, 2(mod 4) ef (0) = n = ef (1)
Then we get, |ef (0)− ef (1)| ≤ 1.
96
5.3. Some New Cycle Related Sum DC Graphs
That is, Wn is SDC graph.
Example 5.3.5. SDC labeling in W5 is demonstrated in the following Figure 5.6.
v1
v2
v3 v4
v5
1
2
4
3 5
6v0
Figure 5.6: SDC labeling in W5
Theorem 5.3.6. The helm Hn is SDC.
Proof. Let V (Hn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is the apex vertex, d(vj) =
4(1 ≤ j ≤ n) and d(uj) = 1(1 ≤ j ≤ n).
Let E(Hn) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}⋃{vjuj | 1 ≤
j ≤ n}.It is to be noted that, |V (Hn)| = 2n+ 1 and |E(Hn)| = 3n.
Consider a bijection f : V (Hn)→ {1, 2, 3, . . . , 2n+ 1} defined as below.
f(v0) = 1.
To label the vertices vj, uj(1 ≤ j ≤ n), let us consider the below possibilities.
Case 1: n ≡ 0, 2(mod 4).
f(v2j−1) = 4j − 1; 1 ≤ j ≤ n
2.
f(v2j) = 4j − 2; 1 ≤ j ≤ n
2.
f(uj) = f(vj) + 2; 1 ≤ j ≤ n.
Case 2: n ≡ 1, 3(mod 4).
f(v2j−1) = 4j − 1; 1 ≤ j ≤ n− 1
2.
f(v2j) = 4j − 2; 1 ≤ j ≤ n− 1
2.
97
5.3. Some New Cycle Related Sum DC Graphs
f(uj) = f(vj) + 2; 1 ≤ j ≤ n− 1.
f(vn) = 2n+ 1.
f(un) = 2n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (0) =⌊3n2
⌋, ef (1) =
⌈3n2
⌉
n ≡ 0, 2(mod 4) ef (0) = 3n2
= ef (1)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Hn is SDC graph.
Example 5.3.6. SDC labeling in H6 is demonstrated in the following Figure 5.7.
1
3
5
7
9
2
4
6
8
11
13
12
14
v3
v4
v5
v1
v2
u3
u4
u5
u6
u1
u2
v0
v6
Figure 5.7: SDC labeling in H6
Theorem 5.3.7. The web graph Wbn is SDC graph.
Proof. Let V (Wbn) = {u0, uj, vj, wj | 1 ≤ j ≤ n}, where u0 is the apex vertex,
uj(1 ≤ j ≤ n) are the vertices corresponding to inner cycle, vj(1 ≤ j ≤ n) are the
vertices corresponding to outer cycle and d(wj) = 1(1 ≤ j ≤ n) are vertices of Wbn.
Let E(Wbn) = {ujuj+1, vjvj+1 | 1 ≤ j ≤ n − 1}⋃{unu1, vnv1}⋃{u0uj, ujvj, vjwj |
1 ≤ j ≤ n}.It is to be noted that, |V (Wbn)| = 3n+ 1 and |E(Wbn)| = 5n.
98
5.3. Some New Cycle Related Sum DC Graphs
Consider a bijection f : V (Wbn)→ {1, 2, 3, . . . , 3n+ 1} defined as below.
f(u0) = 1.
f(uj) = 2j; 1 ≤ j ≤ n.
f(vj) = 2j + 1; 1 ≤ j ≤ n.
f(wj) = (2n+ 1) + j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (0) = 5n2
= ef (1)
n ≡ 1, 3(mod 4) ef (1) =⌊5n2
⌋, ef (0) =
⌈5n2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Wbn is SDC graph.
Example 5.3.7. SDC labeling in Wb5 is demonstrated in the following Figure 5.8.
v1
v2
v3 v4
v5
1
2
4
3
5
6 8
10
79
11
12
13
1415
16
u0
u1
u2
u3
u5
u4
w1
w2
w3 w4
w5
Figure 5.8: SDC labeling in Wb5
Theorem 5.3.8. Shell Sn is SDC graph.
Proof. Let V (Sn) = {vj | 1 ≤ j ≤ n}, where v1 is apex vertex and and vj(2 ≤ j ≤ n)
99
5.3. Some New Cycle Related Sum DC Graphs
are other vertices of shell graph Sn such that
deg(vj) =
2; if j = 2, n.
3; if 3 ≤ j ≤ n− 1.
Let E(Sn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v1vj | 2 ≤ j ≤ n− 1}.
It is to be noted that, |V (Sn)| = n and |E(Sn)| = 2n− 3.
Consider a bijection f : V (Sn)→ {1, 2, 3, . . . , n} defined as below.
Case 1: n ≡ 1, 3(mod 4)
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n− 1.
f(vn) = n.
Case 2: n ≡ 0, 2(mod 4)
f(vj) =
j; j ≡ 1, 2(mod 4).
j + 1; j ≡ 3(mod 4).
j − 1; j ≡ 0(mod 4), 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0, 2, 3(mod 4) ef (1) =⌊2n−3
2
⌋, ef (0) =
⌈2n−3
2
⌉
n ≡ 1(mod 4) ef (1) =⌈2n−3
2
⌉, ef (0) =
⌊2n−3
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Sn is SDC graph.
Example 5.3.8. SDC labeling in S7 is demonstrated in the following Figure 5.9.
100
5.3. Some New Cycle Related Sum DC Graphs
v3
v4 v5
v6
v1
v7
1
2
3
4
5
6
7v2
Figure 5.9: SDC labeling in S7
Theorem 5.3.9. Fln is SDC graph.
Proof. Let V (Fln) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is the apex vertex, d(vj) =
4(1 ≤ i ≤ n) and d(uj) = 2(1 ≤ i ≤ n).
Let E(Fln) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj, vjuj, v0uj | 1 ≤ j ≤ n}.
It is to be noted that, |V (Fln)| = 2n+ 1 and |E(Fln)| = 4n.
Consider a bijection f : V (Fln)→ {1, 2, 3, . . . , 2n+ 1} defined as below.
f(v0) = 1.
f(vj) = 2j; 1 ≤ j ≤ n.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (0) = 2n = ef (1)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Fln is SDC graph.
Example 5.3.9. SDC labeling in Fl4 is demonstrated in the following Figure 5.10.
101
5.3. Some New Cycle Related Sum DC Graphs
u1
u2
u3
u4
v1
v2
v3
v4
1
3
2
5 4
7
6
98v0
Figure 5.10: SDC labeling in Fl4
Theorem 5.3.10. Double fan DFn is SDC graph.
Proof. Let V (DFn) = {u,w, vj | 1 ≤ j ≤ n}, where d(u) = n = d(w) are apex
vertices and uj(1 ≤ j ≤ n) are the vertices on the path Pn corresponding to DFn.
Let E(DFn) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uuj, wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn)| = n+ 2 and |E(DFn)| = 3n− 1.
Consider a bijection f : V (DFn)→ {1, 2, 3, . . . , n+ 2} defined as below.
f(u) = 1.
f(w) = 2.
Case 1: n ≡ 0, 1, 3(mod 4).
f(uj) =
j + 1; j ≡ 3(mod 4)
j + 2; j ≡ 1, 0(mod 4)
j + 3; j ≡ 2(mod 4), 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(uj) =
j + 1; j ≡ 3(mod 4)
j + 2; j ≡ 1, 0(mod 4)
j + 3; j ≡ 2(mod 4), 1 ≤ j ≤ n− 1.
f(un) = n+ 2.
102
5.4. SDC Labeling of Snakes Related Graphs
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (0) = 3n−12
= ef (1)
n ≡ 0, 2(mod 4) ef (1) =⌊3n−1
2
⌋, ef (0) =
⌈3n−1
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DFn is SDC graph.
Example 5.3.10. SDC labeling in DF5 is demonstrated in the following Figure
5.11.
u1 u2 u3 u4 u5
u
w
1
3
2
4 65 7
Figure 5.11: SDC labeling in DF5
5.4 SDC Labeling of Snakes Related Graphs
Vaidya and Shah[52] illuminated some fascinating snakes related graphs for cordial
labeling. Getting inspired by this work, in this section we have emanated some
snakes related graphs for SDC labeling.
Theorem 5.4.1. Triangular snake Tn is SDC graph for n ≡ 0, 1, 2 (mod 4).
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct triangular snake graph Tn from the path Pn, join vj and vj+1 to new
vertex uj by edges e′2j−1 = vjuj and e′2j = vj+1uj, i = 1, 2, . . . , n− 1.
It is to be noted that, |V (Tn)| = 2n− 1 and |E(Tn)| = 3n− 3.
Let us a bijection f : V (Tn)→ {1, 2, 3, . . . , 2n− 1} defined as below.
103
5.4. SDC Labeling of Snakes Related Graphs
Case 1: n ≡ 0, 1(mod 4).
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.
f(ui) = n+ i; 1 ≤ i ≤ n− 1.
Case 2: n ≡ 2(mod 4).
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.
f(u1) = n.
f(uj) = n+ j; 2 ≤ j ≤ n− 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 0(mod 4) ef (1) =⌈3n−3
2
⌉, ef (0) =
⌊3n−3
2
⌋
n ≡ 1(mod 4) ef (1) = 3n−32
= ef (0)
n ≡ 2(mod 4) ef (1) =⌊3n−3
2
⌋, ef (0) =
⌈3n−3
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Tn is SDC graph.
Example 5.4.1. SDC labeling in T6 is demonstrated in the following Figure 5.12.
v1 v2 v3 v4 v5 v6
1
u1 u2 u3 u4 u5
2 4
6 108 11
3 5 7
9
Figure 5.12: SDC labeling in T6
Theorem 5.4.2. Double triangular snake DTn is SDC graph.
104
5.4. SDC Labeling of Snakes Related Graphs
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DTn, join vj and vj+1 to the new vertices uj, wj by edges e′2j−1 =
ujvj, e′2j = ujvj+1, e
′′2j−1 = wjvj, e
′′2j = wjvj+1, j = 1, 2, . . . , n− 1.
It is to be noted that, |V (DTn)| = 3n− 2 and |E(DTn)| = 5n− 5.
Consider a bijection f : V (DTn)→ {1, 2, 3, . . . , 3n− 2} defined as below.
f(vj) =
j; j ≡ 1, 0(mod 4)
j + 1; j ≡ 2(mod 4)
j − 1; j ≡ 3(mod 4) 1 ≤ j ≤ n.
To label the vertices {uj, wj | 1 ≤ j ≤ n− 1}, let us consider the below possibilities.
Case 1: n ≡ 0, 1, 2(mod 4).
f(uj) = n+ j; 1 ≤ j ≤ n− 1.
f(wj) = (2n− 1) + i; 1 ≤ j ≤ n− 1.
Case 2: n ≡ 3(mod 4).
f(u1) = n+ 2.
f(u2) = n+ 1.
f(uj) = n+ j; 3 ≤ j ≤ n− 1.
f(wj) = (2n− 1) + j 1 ≤ j ≤ n− 1.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) = 5n−52
= ef (0)
n ≡ 0(mod 4) ef (1) =⌊5n−5
2
⌋, ef (0) =
⌈5n−5
2
⌉
n ≡ 2(mod 4) ef (1) =⌈5n−5
2
⌉, ef (0) =
⌊5n−5
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DTn is SDC graph.
Example 5.4.2. SDC labeling in DT5 is demonstrated in the following Figure 5.13.
105
5.4. SDC Labeling of Snakes Related Graphs
3
w1 w2 w3 w4
v1v2 v3 v4 v5
1 2 54
u1 u2 u3 u4
6 7 98
10 11 12 13
Figure 5.13: SDC labeling in DT5
Theorem 5.4.3. Quadrilateral snake Qn is SDC graph.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct Qn, join vj and vj+1 to the new vertices uj, wj by edges e′2j−1 =
vjuj, e′2j = vj+1wj and e′′j = ujwj, j = 1, 2, . . . n− 1.
It is to be noted that, |V (Qn)| = 3n− 2 and |E(Qn)| = 4n− 4.
Consider a bijection f : V (Qn)→ {1, 2, 3, . . . , 3n− 2} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.
To label the vertices {uj, wj | 1 ≤ j ≤ n− 1}, let us consider the below possibilities.
Case 1: n ≡ 0(mod 4).
f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.
Case 2: n ≡ 2(mod 4).
f(u1) = n,
f(uj) = n+ 2j − 1; 2 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.
106
5.4. SDC Labeling of Snakes Related Graphs
In above two cases, whenever j ≡ 0(mod 4) interchange f(uj) and f(wj).
Case 3: n ≡ 1(mod 4).
f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.
Case 4: n ≡ 3(mod 4).
f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.
In above two cases, whenever j ≡ 2(mod 4) interchange f(uj) and f(wj)
By looking into the above prescribed pattern,
ef (1) = 2n− 2 = ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Qn is SDC graph.
Example 5.4.3. SDC labeling in Q5 is demonstrated in the following Figure 5.14.
11
2 4
6 108 12
1 3 5
7 9
v1 v2 v3 v4 v5
w1 w2u1 u2 w3u3 w4u4
13
Figure 5.14: SDC labeling in Q5
Theorem 5.4.4. Double quadrilateral snake DQn is SDC graph.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DQn, join vj and vj+1 to new vertices uj, u
′j, wj and w′j by edges
e(u)2j−1 = viuj, e
(u)2j = vj+1u
′j, e
(uu)j = uju
′j, e
(w)2j−1 = vjwj, e
(w)2j = vj+1w
′j, e
(ww)j = wjw
′j
for j = 1, 2, . . . , n− 1.
107
5.4. SDC Labeling of Snakes Related Graphs
It is to be noted that, |V (DQn)| = 5n− 4 and |E(DQn)| = 7n− 7.
Consider a bijection f : V (DQn)→ {1, 2, 3, . . . , 5n− 4} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.
To label the vertices {uj, u′j | 1 ≤ j ≤ n − 1} and {wj, w′j | 1 ≤ j ≤ n − 1}, let us
consider the below possibilities.
Case 1: n ≡ 0, 1, 3(mod 4).
f(uj) = n+ 2j − 1; 1 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 1 ≤ j ≤ n− 1.
f(u′j) = 3n− 3 + 2j; 1 ≤ j ≤ n− 1.
f(w′j) = 3n− 2 + 2j; 1 ≤ j ≤ n− 1.
Case 2: n ≡ 2(mod 4).
f(u1) = n, f(u′1) = n+ 2,
f(uj) = n+ 2j − 1; 2 ≤ j ≤ n− 1.
f(wj) = n+ 2j; 2 ≤ j ≤ n− 1.
f(u′j) = 3n− 3 + 2j; 1 ≤ j ≤ n− 1.
f(w′j) = 3n− 2 + 2j; 1 ≤ j ≤ n− 1.
For n ≡ 0, 2(mod 4) : whenever j ≡ 0(mod 4) interchange f(uj) with f(wj) and
f(u′j) with f(w′j).
For n ≡ 1, 3(mod 4) : whenever j ≡ 2(mod 4) interchange f(uj) with f(wj) and
f(u′j) with f(w′j).
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) = 7(n−1)2
= ef (0)
n ≡ 0, 2(mod 4) ef (1) =⌊7(n−1)
2
⌋, ef (0) =
⌈7(n−1)
2
⌉
108
5.4. SDC Labeling of Snakes Related Graphs
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DQn is SDC graph.
Example 5.4.4. SDC labeling in DQ5 is demonstrated in the following Figure 5.15.
v1 v2 v3 v4 v5
w1 w2
2
u1 u2 w3u3 w4u4
2 41 3 5
116 108 127 9 13
14 15 17 16 18 19 20 21
w'1 w'2u'1 u'2 w'3u'3 w'4u'4
Figure 5.15: SDC labeling in DQ5
Theorem 5.4.5. Alternate triangular snake A(Tn) confesss SDC labeling.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct A(Tn), join v2i−1 and v2i to new vertex ui, where 1 ≤ i ≤ bn
2c.
Therefore V (A(Tn)) = {vi, uj/1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.
It is to be noted that
|V (A(Tn))| =
3n2
; if n is even.
3n−12
; if n is odd.
and
|E(A(Tn))| =
2n− 1 ; if n is even.
2n− 2 ; if n is odd.
Consider a bijection f : V (A(Tn))→ {1, 2, 3, . . . , |V (A(Tn))|} defined as below.
f(vi) =
i+ 2 ; i ≡ 1(mod 4)
i ; i ≡ 2(mod 4)
i+ 1 ; i ≡ 0, 3(mod 4) 1 ≤ i ≤ n.
f(u1) = 1.
109
5.4. SDC Labeling of Snakes Related Graphs
Case 1: n ≡ 0, 2, 3(mod 4).
f(ui) = n+ i; 2 ≤ i ≤⌊n
2
⌋.
Case 2: n ≡ 1(mod 4).
f(u2) = n+ 1,
f(ui) = n+ i; 3 ≤ i ≤ n− 1
2.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) = n− 1 = ef (0)
n ≡ 0, 2(mod 4) ef (1) =⌊2n−1
2
⌋, ef (0) =
⌈2n−1
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, A(Tn) is SDC graph.
Example 5.4.5. SDC labeling in A(T7) is demonstrated in the following Figure
5.16.
u1 u2 u3
v1 v2 v3 v4 v5 v6 v7
3
1
2 6754 8
9 10
Figure 5.16: SDC labeling in A(T7)
Theorem 5.4.6. Alternate quadrilateral snake A(Qn) confesss SDC labeling.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.Then A(Qn) will be constructed by joining v2i−1 and v2i to new vertices ui and wi
respectively and then joining ui and wi, where 1 ≤ i ≤ bn2c.
Therefore V (A(Qn)) = {vi, uj, wj/1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.
110
5.4. SDC Labeling of Snakes Related Graphs
It is to be noted that
|V (A(Qn))| =
2n; if n is even.
2n− 1; if n is odd.
and
|E(A(Qn))| =
5n−22
; if n is even.
5n−52
; if n is odd.
Consider a bijection f : V (A(Qn))→ {1, 2, 3, . . . , |V (A(Qn))|} defined as below.
Case 1: n ≡ 0, 2(mod 4).
f(vi) =
2i− 1; i ≡ 1, 2, 3, 4(mod 8)
2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n.
f(ui) =
4i− 2; i ≡ 1, 2, (mod 4)
4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n2.
f(wi) =
4i; ; i ≡ 1, 2, (mod 4)
4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n2.
Case 2: n ≡ 1, 3(mod 4).
f(vi) =
2i− 1 ; i ≡ 1, 2, 3, 4(mod 8)
2i ; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n− 1.
f(vn) = 2n− 1.
f(ui) =
4i− 2; i ≡ 1, 2, (mod 4)
4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n−12.
f(wi) =
4i; i ≡ 1, 2, (mod 4)
4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤ n−12.
By looking into the above prescribed pattern,
111
5.4. SDC Labeling of Snakes Related Graphs
Cases of n Edge conditions
n ≡ 1(mod 4) ef (1) = 5n−52
= ef (0)
n ≡ 3(mod 4) ef (1) =⌊5n−5
2
⌋, ef (0) =
⌈5n−5
2
⌉
n ≡ 2(mod 4) ef (1) = 5n−22
= ef (0)
n ≡ 0(mod 4) ef (1) =⌈5n−2
2
⌉, ef (0) =
⌊5n−2
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, A(Qn) is SDC graph.
Example 5.4.6. SDC labeling in A(Q8) is demonstrated in the following Figure
5.17.
v1 v2 v3 v4 v5 v6 v7 v8
w1 w2 w3 w4u1 u2 u3 u4
1
2 4
3 5 7
6
10
8 119
12
13
14
15
16
Figure 5.17: SDC labeling in A(Q8)
Theorem 5.4.7. Double alternate triangular snake DA(Tn) confesss SDC labeling.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DA(Tn), join v2i−1 and v2i to new vertices ui and wi respectively,
1 ≤ i ≤ bn2c.
Then V (DA(Tn))) = {vi, uj, wj | 1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.
It is to be noted that
|V (DA(Tn))| =
2n; if n is even.
2n− 1; if n is odd.
and
|E(DA(Tn))| =
3n− 1; if n is even.
3n− 3; if n is odd.
112
5.4. SDC Labeling of Snakes Related Graphs
Consider a bijection f : V (DA(Tn))→ {1, 2, 3, . . . , |V (DA(Tn))|} defined as below.
f(vi) =
i+ 2; i ≡ 1(mod 4)
i; i ≡ 2(mod 4)
i+ 1; i ≡ 0, 3(mod 4) 1 ≤ i ≤ n.
f(u1) = 1.
To label the vertices {ui | 2 ≤ i ≤ n − 1} and {wi | 1 ≤ i ≤ n − 1}, consider the
labeling defined by means of the below cases.
Case 1: n ≡ 0, 2(mod 4).
f(ui) = n+ i; 2 ≤ i ≤ n
2.
f(wi) =3n
2+ i; 1 ≤ i ≤ n
2.
Case 2: n ≡ 1(mod 4).
f(u2) = n+ 1.
f(ui) = n+ i; 3 ≤ i ≤ n− 1
2,
f(wi) =3(n− 1)
2+ 1 + i; 1 ≤ i ≤ n− 1
2.
Case 3: n ≡ 3(mod 4).
f(ui) = n+ i; 2 ≤ i ≤ n− 1
2,
f(wi) =3(n− 1)
2+ 1 + i; 1 ≤ i ≤ n− 1
2.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) = 3n−32
= ef (0)
n ≡ 0, 2(mod 4) ef (1) =⌊3n−1
2
⌋, ef (0) =
⌈3n−1
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DA(Tn) is SDC graph.
113
5.4. SDC Labeling of Snakes Related Graphs
Example 5.4.7. SDC labeling in DA(T10) is demonstrated in the following Figure
5.18.
u1 u2 u3 u4 u5
1
2 4v1 v2 v3 v4 v5 v6 v7 v8 v9 v10
3 675 98 1011
13 14 1512
17 201816 19
w1 w2 w3 w4 w5
Figure 5.18: SDC labeling in DA(T10)
Theorem 5.4.8. Double alternate quadrilateral snake DA(Qn) confesss SDC label-
ing.
Proof. Let Pn be a path with V (Pn) = {vj | 1 ≤ j ≤ n} and E(Pn) = {vjvj+1 | 1 ≤j ≤ n− 1}.To construct DA(Qn), join v2i−1 and v2i to new vertices ui, wi and u′i, w
′i respectively
and adding the edges uiwi and u′iw′i, 1 ≤ i ≤ bn
2c.
Then V (DA(Qn)) = {vi, uj, wj, u′j, w′j | 1 ≤ i ≤ n, 1 ≤ j ≤ bn2c}.
It is to be noted that
|V (DA(Qn))| =
3n; if n is even.
3n− 2; if n is odd.
and
|E(DA(Qn))| =
4n− 1; if n is even.
4n− 4; if n is odd.
Consider a bijection f : V (DA(Qn))→ {1, 2, 3, . . . , |V (DA(Qn))|} defined as below.
Case 1: n ≡ 0, 2(mod 4).
f(vi) =
2i− 1; i ≡ 1, 2, 3, 4(mod 8)
2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n.
114
5.4. SDC Labeling of Snakes Related Graphs
f(u′1) = 2n+ 1.
f(u′2) = 2n+ 2.
f(u′i) = 2n+ 2i− 1; 3 ≤ i ≤⌊n
2
⌋.
f(w′1) = 2n+ 3,
f(w′i) = 2n+ 2i; 2 ≤ i ≤⌊n
2
⌋.
Case 2: n ≡ 1, 3(mod 4).
f(vi) =
2i− 1; i ≡ 1, 2, 3, 4(mod 8)
2i; i ≡ 0, 5, 6, 7(mod 8) 1 ≤ i ≤ n− 1.
f(vn) = 3n− 2.
f(u′1) = 2n− 1.
f(u′2) = 2n.
f(u′i) = 2(n− 1) + 2i− 1; 3 ≤ i ≤⌊n
2
⌋.
f(w′1) = 2n+ 1.
f(w′i) = 2(n− 1) + 2i; 2 ≤ i ≤⌊n
2
⌋.
For n ≡ 0, 1, 2, 3(mod 4).
f(ui) =
4i− 2; i ≡ 1, 2, (mod 4)
4i− 3; i ≡ 3, 4(mod 4) 1 ≤ i ≤⌊n2
⌋.
f(wi) =
4i; i ≡ 1, 2, (mod 4)
4i− 1; i ≡ 3, 4(mod 4) 1 ≤ i ≤⌊n2
⌋.
By looking into the above prescribed pattern,
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) = 2n− 2 = ef (0)
n ≡ 0, 2(mod 4) ef (1) =⌊4n−1
2
⌋, ef (0) =
⌈4n−1
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DA(Qn) is SDC graph.
115
5.5. Conclusion and Scope for Further Research
Example 5.4.8. SDC labeling in DA(Q9) is demonstrated in the following Figure
5.19.
w1 w2 w3 w4u1 u2 u3 u4
v1v2 v3 v4 v5 v6 v7 v8
w'1 w'2 w'3 w'4u'1 u'2 u'3 u'4
1
2 4
3 5 7
6
10
8 119
12
13
14
15
16
17 1819 20 21 22 23 24
25v9
Figure 5.19: SDC labeling in DA(Q9)
5.5 Conclusion and Scope for Further Research
In this chapter we have emanated some new standard SDC graphs and snake related
SDC graphs.
However, there is no such standard relation between either of the two DC and SDC
labelings; like, we may find a graph which confess one labeling but not the other.
To observe this matter more effectively, the list of graph families satisfying/ not
satisfying certain labeling is shown below.
• The star K1,n is both DC and SDC (Refer [44] and [1]).
• The triangular snake graph DT5 is not DC (easy to check) but it is SDC (Refer
Theorem 5.4.2).
• The triangular snake graph T7 is neither DC (easy to check) nor SDC (Refer
Theorem 5.4.1).
At the end, we give some problems.
Problem 5.5.1. Derive essential and adequate condition (if any) for any graph to
be SDC graph.
Problem 5.5.2. To construct similar results for other graph families and to make
the relation stonger between the two labeling mentioned.
116
5.5. Conclusion and Scope for Further Research
The penultimate chapter is also based on SDC graphs where SDC labeling is going
to be discussed for the graphs obtained by using graph operations.
117
CHAPTER 6
Sum Divisor Cordial Labeling
With the Use of Some Graph
Operations
SDC labeling have been discussed for certain graph families in the earlier Chapter-5,
while the existing chapter aims to give a brief account of SDC labeling in the graphs
constructed by using the following graph operations.
z Ringsum of different graphs with star graph K1,n.
z Corona of different graphs with graph K1.
z Vertex switching of graphs.
z Duplication a vertex in star, cycle and path allied graphs.
6.1 SDC Labeling of Graphs With the Use of Ringsum of
Different Graphs with Star Graph K1,n
Ghodasara and Rokad[5] illuminated and derived some fascinating results on cordial
labeling of the graphs by considering ringsum of K1,n with different graph families.
Under the inspiration of this credibility, in the current segment we demonstrate
some results on SDC labeling of the graphs constructed from the graph operation
ringsum.
118
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Remark 6.1.1. Throughout this chapter we consider the ringsum of a graph G with
K1,n by considering any one vertex of G and apex vertex of K1,n as a common vertex.
Theorem 6.1.1. Cn ⊕K1,n is SDC.
Proof. Let V (Cn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Cn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}.
It is to be noted that, |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.
Consider a bijection f from V (Cn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(uj) = 2j − 1; 1 ≤ j ≤ n.
f(vk) = 2k; 1 ≤ k ≤ n.
As per this pattern, allocate the vertex labels such that for any edge uiui+1 ∈E(Cn ⊕K1,n),
f(ui) | f(ui+1), 1 ≤ i ≤ n− 1
and
f(u1) - f(vj) 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = ef (0) = n.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn ⊕K1,n is SDC graph.
Example 6.1.1. SDC labeling in the graph C5⊕K1,5 is demonstrated in the following
Figure 6.1.
119
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
1
v3 v5v1 v2 v4
3
2 4
5
6
7
8
9
10
u1
u2
u3
u5
u4
Figure 6.1: SDC labeling in C5 ⊕K1,5
Theorem 6.1.2. G⊕K1,n is sum DC graph, where G is cycle Cn with one chord.
Proof. Let cycle Cn with one chord be denoted as G.
Let V (G ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = V (Cn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u2un},
where u2un is the chord of Cn and vertices u1, u2, un form a triangle with chord u2un.
Also it is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 1.
Consider a bijection f from V (G⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(uj) = 2j − 1; 1 ≤ j ≤ n.
f(vk) = 2k; 1 ≤ k ≤ n.
As per this pattern, allocate the vertex labels such that for any edge ujuj+1 ∈E(G⊕K1,n),
f(uj) | f(uj+1) 1 ≤ j ≤ n− 1
and f(u1) - f(vk) 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (0) = n, ef (1) = n+ 1.
Then we get, |ef (0)− ef (1)| ≤ 1.
120
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
That is, G⊕K1,n is SDC graph, where G is cycle Cn with one chord.
Example 6.1.2. SDC labeling in the graph constructed from ringsum of C7 with
one chord and K1,7 is demonstrated in the following Figure 6.2.
u3
u4 u5
u6
u7
u1
u2
v3 v4 v5 v6 v7v1 v2
5
7
3
1
9
2 4 6 8 10 12 14
11
13
Figure 6.2: SDC labeling in the graph constructed from ringsum of C7 with one chord and K1,7
Theorem 6.1.3. Cn,3 ⊕K1,n is a sum DC graph.
Proof. Let V (Cn,3 ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Cn,3) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(Cn,3 ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤
n}⋃{u2un, u2un−1}, where u2un and u2un−1 are the chords of Cn.
It is to be noted that, |V (Cn,3 ⊕K1,n)| = 2n and |E(Cn,3 ⊕K1,n)| = 2n+ 2.
Consider a bijection f from V (Cn,3 ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(uj) = 2j − 1; 1 ≤ j ≤ n− 3.
f(un−2) = 2n.
f(uj) = 2j − 3; n− 1 ≤ j ≤ n.
f(vk) = 2k; 1 ≤ k ≤ n− 1.
f(vn) = 2n− 1.
As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(Cn,3 ⊕K1,n),
f(uj) | f(uj+1), 1 ≤ j ≤ n− 4.
Also f(u1) - f(vk), 1 ≤ k ≤ n− 1.
121
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
By looking into the above prescribed pattern,
ef (0) = n+ 1 = ef (1).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn,3 ⊕K1,n is SDC graph.
Example 6.1.3. SDC labeling in the graph C7,3 ⊕K1,7 is demonstrated in the fol-
lowing Figure 6.3.
2
11
u3
u4 u5
u6
u7
u1
u2
v3 v4 v5 v6 v7v1 v2
3
5
7
9
13
1
4 6 8 10 12
14
Figure 6.3: SDC labeling in C7,3 ⊕K1,7
Theorem 6.1.4. Cn(1, 1, n− 5)⊕K1,n is SDC graph.
Proof. Let G be the cycle with triangle Cn(1, 1, n− 5).
Let V (G ⊕ K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (G) = {uj | 1 ≤ j ≤ n} and
V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(G⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u1vj | 1 ≤ j ≤ n}⋃{u1u3,
u3un−1, un−1u1}, where u1, u3 and un−1 are the vertices of the triangle formed by the
chords u1u3, u3un−1 and u1un−1.
It is to be noted that, |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 3.
Consider a bijection f from V (G⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 1.
f(uj) = 2j − 1; 2 ≤ j ≤ n− 1.
122
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
f(un) = 2n.
f(vk) = 2k; 1 ≤ k ≤ n− 1.
f(vn) = 2n− 1.
As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(G⊕K1,n),
f(uj) | f(uj+1), 1 ≤ j ≤ n− 2.
f(u1) - f(vk), 1 ≤ k ≤ n− 1
f(u1) | f(u3), f(u3) | f(un−1), f(un−1) | f(u1).
By looking into the above prescribed pattern,
ef (0) = n+ 1, ef (1) = n+ 2.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn(1, 1, n− 5)⊕K1,n is SDC graph.
Example 6.1.4. SDC labeling in the graph C8(1, 1, 3)⊕K1,8 is demonstrated in the
following Figure 6.4.
v3 v4 v5 v6 v7v1 v2
u3
u4
u5
u6
u7
u1
v8
u2 u8
2
1
5
7
9
11
13
15
3
4 6 8 10 12
16
14
Figure 6.4: SDC labeling in C8(1, 1, 3)⊕K1,8
Theorem 6.1.5. Wn ⊕K1,n is SDC graph.
Proof. Let V (Wn ⊕K1,n) = {u0, uj, vj | 1 ≤ j ≤ n}, where V (Wn) = {u0, uj | 1 ≤j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.
123
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Here u0 is apex vertex, uj(1 ≤ j ≤ n) are rim vertices of Wn and vj(1 ≤ j ≤ n) are
the pendant vertices, u1 is apex vertex of star graph.
Let E(Wn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n−1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u1vj |
1 ≤ j ≤ n}.It is to be noted that, |V (Wn ⊕K1,n)| = 2n+ 1 and |E(Wn ⊕K1,n)| = 3n.
Consider a bijection f from V (Wn ⊕K1,n) to {1, 2, 3, . . . , 2n+ 1} defined as below.
f(u0) = 2.
f(u1) = 1.
f(uj) =
j ; j ≡ 0(mod 4)
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4) 2 ≤ j ≤ n.
f(vk) = f(un) + 1 + k; 1 ≤ k ≤ n.
By looking into the above prescribed pattern
ef (1) =
⌈3n
2
⌉, ef (0) =
⌊3n
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Wn ⊕K1,n is SDC graph.
Example 6.1.5. SDC labeling in the graph W6 ⊕ K1,6 is demonstrated in the fol-
lowing Figure 6.5.
9
5
7
11 13
1
3
4
6
2
8 10 12
u0
v6
u3
u4
u5
u1
u2
v1 v2 v4v3 v5
u6
Figure 6.5: SDC labeling in W6 ⊕K1,6
124
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Theorem 6.1.6. Fln ⊕K1,n is SDC graph.
Proof. Let V (Fln⊕K1,n) = {u0, uj, vj, wj | 1 ≤ j ≤ n}, where V (Fln) = {u0, uj, wj |1 ≤ j ≤ n} and V (K1,n) = {w1, vj | 1 ≤ j ≤ n}.Here u0 is apex vertex, uj(1 ≤ j ≤ n) are internal vertices and wj(1 ≤ j ≤ n) are
external vertices of Fln and d(vj) = 1, where 1 ≤ j ≤ n, w1 is apex vertex of star
graph.
Let E(Fln ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ n − 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤
n}⋃{u0wj | 1 ≤ j ≤ n}⋃{ujwj | 1 ≤ j ≤ n}⋃{w1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Fln ⊕K1,n)| = 3n+ 1 and |E(Fln ⊕K1,n)| = 5n.
Consider a bijection f from V (Fln ⊕K1,n) to {1, 2, 3, . . . , 3n+ 1} defined as below.
f(u0) = 1.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
f(wj) = 2j; 1 ≤ j ≤ n.
f(vk) = f(un) + k; 1 ≤ k ≤ n.
As per this pattern, allocate the vertices such that for any edge ujuj+1 ∈ E(Fln ⊕K1,n),
f(uj) | f(uj+1), 1 ≤ j ≤ n− 1.
Further f(u0) | f(uj), f(u0) - f(wj), 1 ≤ j ≤ n
and f(w1) | f(vk) whenever k is odd, 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌈5n
2
⌉, ef (0) =
⌊5n
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Fln ⊕K1,n is SDC graph.
Example 6.1.6. SDC labeling in the graph Fl4 ⊕ K1,4 is demonstrated in the fol-
lowing Figure 6.6.
125
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
3
2
9 854
7
6
11
1310
12
1
u1
u2
u3
u4
w1
w2
w3
u0
v1
v2 v3
v4
w4
Figure 6.6: SDC labeling in Fl4 ⊕K1,4
Theorem 6.1.7. Gn ⊕K1,n is SDC graph.
Proof. Let V (Gn ⊕ K1,n) = {u0, uj, vk | 1 ≤ j ≤ 2n, 1 ≤ k ≤ n}, where V (Gn) =
{u0, uj | 1 ≤ j ≤ 2n} and V (K1,n) = {u1, vk | 1 ≤ k ≤ n}.Here u0 is apex vertex, d(u2i−1) = 3(1 ≤ i ≤ n) and d(u2i) = 2(1 ≤ i ≤ n) and
d(vk) = 1, where 1 ≤ k ≤ n, u1 is apex vertex of star graph.
Let E(Gn ⊕ K1,n) = {ujuj+1 | 1 ≤ j ≤ 2n − 1}⋃{u2nu1}⋃{u0u2j−1 | 1 ≤ j ≤
n}⋃{u1vk | 1 ≤ k ≤ n}.It is to be noted that, |V (Gn ⊕K1,n)| = 3n+ 1 and |E(Gn ⊕K1,n)| = 4n.
Consider a bijection f from V (Gn ⊕K1,n) to {1, 2, 3, . . . , 3n+ 1} define as belows.
Case:1 n ≡ 1, 3(mod 4)
f(u0) = 2.
f(u1) = 1.
f(uj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4) 2 ≤ j ≤ n.
f(vk) = f(un−1) + k; 1 ≤ k ≤ n.
126
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Case:2 n ≡ 2, 4(mod 4)
f(u0) = 2.
f(u1) = 1.
f(uj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4) 2 ≤ j ≤ n− 1.
f(un) = 3n+ 1.
f(vk) = f(un−1) + k; 1 ≤ k ≤ n− 1.
f(vn) = 2n.
By looking into the above prescribed pattern, ef (1) = ef (0) = 2n.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Gn ⊕K1,n is SDC graph.
Example 6.1.7. SDC labeling in the graph G6⊕K1,6 is demonstrated in the following
Figure 6.7.
u0
u2
u3
u4
u5
u6
u7
u8
u9
u10
u12
u1
u11
7
v1 v2 v3 v4 v5 v6
1
2
35
4
6
98
10
11
13
19
18171614 15 12
Figure 6.7: SDC labeling in G6 ⊕K1,6
Theorem 6.1.8. Pn ⊕K1,n is SDC graph.
Proof. Let V (Pn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Pn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
127
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Let E(Pn ⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{u1vj | 1 ≤ j ≤ n}.It is to be noted that, |V (Pn ⊕K1,n)| = 2n and |E(Pn ⊕K1,n)| = 2n− 1.
Consider a bijection f from V (Pn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(uj) = 2j; 1 ≤ j ≤ n.
f(vk) = 2k − 1; 1 ≤ k ≤ n.
As per this pattern, allocate the vertices such that for any edge uiui+1 ∈ E(Pn ⊕K1,n),
f(uj) | f(uj+1), 1 ≤ j ≤ n− 1.
Also f(u1) - f(vk) 1 ≤ k ≤ n.
By looking into the above prescribed pattern
ef (0) = n, ef (1) = n− 1.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Pn ⊕K1,n is SDC graph.
Example 6.1.8. SDC labeling in the graph P5⊕K1,5 is demonstrated in the following
Figure 6.8.
246810u1u2u3u4u5
v1
v2
v3
v4
v5
5
7
31
9
Figure 6.8: SDC labeling in P5 ⊕K1,5
Theorem 6.1.9. Sn ⊕K1,n is SDC graph.
Proof. Let V (Sn ⊕K1,n) = {uj, vj | 1 ≤ j ≤ n}, where V (Sn) = {uj | 1 ≤ j ≤ n}and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of Sn as well as of star graph.
Let E(Sn⊕K1,n) = {ujuj+1, unu1 | 1 ≤ j ≤ n−1}⋃{u1uj | 2 ≤ j ≤ n−1}⋃{u1vj |
128
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
1 ≤ j ≤ n}.It is to be noted that, |V (Sn ⊕K1,n)| = 2n and |E(Sn ⊕K1,n)| = 3n− 3.
Consider a bijection f from V (Sn ⊕K1,n) to {1, 2, 3, . . . , 2n} defined as below.
f(u1) = 1.
f(uj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4) 2 ≤ j ≤ n.
To label the vertices {vj | 1 ≤ j ≤ n}, let us consider the below possibilities.
Case:1 n ≡ 0, 1, 2(mod 4).
f(vk) = n+ k; 1 ≤ k ≤ n.
Case:2 n ≡ 3(mod 4).
f(v1) = n
f(vk) = n+ k; 2 ≤ k ≤ n.
The edge label conditions constructed due to the above labeling pattern is shown in
the below table.
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) =⌈3n−3
2
⌉, ef (1) =
⌊3n−3
2
⌋
n ≡ 1, 3(mod 4) ef (1) = 3n−32
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Sn ⊕K1,n is SDC graph.
Example 6.1.9. SDC labeling in the graph S7⊕K1,7 is demonstrated in the following
Figure 6.9.
129
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
u3
u4u5
u6
u7
u1
u2
v3 v4 v5 v6 v7v1 v2
3 5
7 9 13
1
4 6
8
10 12 14
2
11
Figure 6.9: DC labeling in S7 ⊕K1,7
Theorem 6.1.10. DFn ⊕K1,n is SDC graph.
Proof. Let V (DFn ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (DFn) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u, vj | 1 ≤ j ≤ n}.Here u,w are apex vertices of DFn, d(uj) = 3(1 ≤ j ≤ n), d(vj) = 1(1 ≤ j ≤ n) and
u is apex vertex of star graph.
Let E(DFn⊕K1,n) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{uvj | 1 ≤ j ≤ n}⋃{uuj | 1 ≤ j ≤n}⋃{wuj | 1 ≤ j ≤ n}.It is to be noted that, |V (DFn ⊕K1,n)| = 2n+ 2 and |E(DFn ⊕K1,n)| = 4n− 1.
Consider a bijectionf from V (DFn⊕K1,n) to {1, 2, 3, . . . , 2n+ 2} defined as below.
f(w) = 2.
f(u) = 1.
f(uj) =
j + 2 ; j ≡ 0, 1(mod 4)
j + 3 ; j ≡ 2(mod 4)
j + 1 ; j ≡ 3(mod 4) 1 ≤ j ≤ n.
To label the vertices {vj | 1 ≤ j ≤ n}, let us consider the below possibilities.
Case:1 n ≡ 0, 1, 3(mod 4).
f(vk) = f(un) + k; 1 ≤ k ≤ n.
130
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
Case:2 n ≡ 2(mod 4).
f(v1) = n+ 2.
f(vk) = f(un) + k; 2 ≤ k ≤ n.
The edge label conditions constructed due to the above labeling pattern is shown in
the below table.
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) = 2n, ef (1) = 2n− 1
n ≡ 1, 3(mod 4) ef (1) = 2n, ef (0) = 2n− 1
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DFn ⊕K1,n is SDC graph.
Example 6.1.10. SDC labeling in the graph DF5 ⊕ K1,5 is demonstrated in the
following Figure 6.10.
u1 u2 u3 u4 u5
7
9
v1
v2
v3
v4
v5
u
w
1
3 5
11
2
4 6
8
10
12
v1
v2
v3
v4
v5
u
w
1
3 5
11
2
4 6
8
10
12
u1 u2 u3 u4 u5
7
9
Figure 6.10: SDC labeling in DF5 ⊕K1,5
Theorem 6.1.11. K2,n ⊕K1,n is SDC graph.
Proof. Let V (K2,n ⊕K1,n) = {u,w, uj, vj | 1 ≤ j ≤ n}, where V (K2,n) = {u,w, uj |1 ≤ j ≤ n} and V (K1,n) = {u1, vj | 1 ≤ j ≤ n}.Here d(vj) = 1, where 1 ≤ j ≤ n and u1 is apex vertex of star graph.
Let E(K2,n ⊕K1,n) = {uuj | 1 ≤ j ≤ n}⋃{wuj | 1 ≤ j ≤ n}⋃{u1vj | 1 ≤ j ≤ n}.
131
6.1. SDC Labeling of Graphs With the Use of Ringsum of Different Graphs with Star Graph K1,n
It is to be noted that, |V (K2,n ⊕K1,n)| = 2n+ 2 and |E(K2,n ⊕K1,n)| = 3n.
Consider a bijection f from V (K2,n⊕K1,n) to {1, 2, 3, . . . , 2n+ 2} defined as below.
f(u) = 1.
f(w) = 2.
f(uj) = j + 2; 1 ≤ j ≤ n.
f(vk) = f(un) + k; 1 ≤ k ≤ n.
The edge label conditions constructed due to the above labeling pattern is shown in
the below table.
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) = 3n2
= ef (1)
n ≡ 1, 3(mod 4) ef (0) =⌈3n2
⌉, ef (1) =
⌊3n2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K2,n ⊕K1,n is SDC graph.
Example 6.1.11. SDC labeling in the graph K2,7 ⊕ K1,7 is demonstrated in the
following Figure 6.11.
u1
u2
v1
v2
u3
v3 v4
v5
u4
w
u
u5
u6
u7
v6
v7
1 5
7
1113
9
15
3
2
4
6
8
10
1214
16
Figure 6.11: SDC labeling in K2,7 ⊕K1,7
132
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
6.2 SDC Labeling in the Graphs constructed from Corona
Product with K1
Kanani and Bosmia[25] illuminated and derived some fascinating graphs by consid-
ering corona product of K1 with different graph families for divisor cordial labeling.
Under the inspiration of this credibility, in the current segment we demonstrate some
new graphs constructed from the graph operation corona product for SDC labeling.
Theorem 6.2.1. K1,n �K1 is SDC graph.
Proof. Let V (K1,n�K1) = {vj, v′j | 0 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤n) are pendant vertices of K1,n and v′j(0 ≤ j ≤ n) are the lately inserted vertices to
construct the graph K1,n �K1.
Let E(K1,n �K1) = {v0vj; 1 ≤ j ≤ n}⋃{vjv′j; 0 ≤ j ≤ n}.Also it is to be noted that, |V (K1,n �K1)| = 2n+ 2 and |E(K1,n �K1)| = 2n+ 1.
Consider a bijection f : V (K1,n �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.
f(v0) = 1.
f(v′0) = 2.
f(vj) = 2j + 1 1 ≤ j ≤ n.
f(v′j) = 2j + 2 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌊2n+ 1
2
⌋, ef (0) =
⌈2n+ 1
2
⌉.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K1,n �K1 is SDC graph.
Example 6.2.1. SDC labeling in the graph K1,6 � K1 is demonstrated in the fol-
lowing Figure 6.12.
133
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
v1 v2 v3 v4 v5 v6
v2' v3' v4' v5' v6'
v0'
v01
6
3
4 8 1210 14
5 1197 13
2
v1'
Figure 6.12: SDC labeling in K1,6 �K1
Theorem 6.2.2. K2,n �K1 is sum divisor cordial graph for n ≡ 1, 2, 3(mod 4).
Proof. Let V (K2,n � K1) = {uj, u′j | 1 ≤ j ≤ 2}⋃{vj, v′j, | 1 ≤ j ≤ n}, where
V (K2,n) = {u1, u2, vj | 1 ≤ j ≤ n} and u′1, u′2, v′j(1 ≤ j ≤ n) are lately inserted
vertices to construct the graph K2,n �K1.
Let E(K2,n �K1) = {u1vj, u2vj | 1 ≤ j ≤ n}⋃{u1u′1, u2u′2}⋃{vjv′j; 1 ≤ j ≤ n}.
Also it is to be noted that, |V (K2,n �K1)| = 2n+ 4 and |E(K2,n �K1)| = 3n+ 2.
Consider a bijection f : V (K2,n �K1)→ {1, 2, 3, . . . , 2n+ 4} defined as below.
f(u1) = 1.
f(u2) = 2.
f(vj) = j + 2; 1 ≤ j ≤ n.
Case 1: n ≡ 1(mod 4).
f(u′1) = 2n+ 3.
f(u′2) = 2n+ 4.
f(v′2j−1) = (n+ 2) + 2j 1 ≤ j ≤ k.
f(v′2j) = (n+ 2) + (2j − 1); 1 ≤ j ≤ k.
f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n− 1
4.
Case 2: n ≡ 2(mod 4).
f(u′1) = 2n+ 3.
134
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
f(u′2) = 2n+ 4.
f(v′2j−1) = (n+ 2) + (2j); 1 ≤ j ≤ k.
f(v′2j) = (n+ 2) + (2j − 1); 1 ≤ j ≤ k.
f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n+ 2
4.
Case 3: n ≡ 3(mod 4).
f(u′1) = 2n+ 4.
f(u′2) = 2n+ 3.
f(v′2j−1) = (n+ 2) + 2j 1 ≤ j ≤ k.
f(v′2j) = (n+ 1) + (2j − 1); 1 ≤ j ≤ k.
f(v′j) = (n+ 2) + j; 2k + 1 ≤ j ≤ n,where k =n+ 1
4.
By looking into the above prescribed pattern, the below table describes edge label
conditions.
Cases of n Edge label conditions
n ≡ 1(mod 4) ef (1) =⌈3n+2
2
⌉, ef (0) =
⌊3n+2
2
⌋
n ≡ 2(mod 4) ef (1) = 3n+22
= ef (0)
n ≡ 3(mod 4) ef (1) =⌊3n+2
2
⌋, ef (0) =
⌈3n+2
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K2,n �K1 is SDC graph.
Example 6.2.2. SDC labeling in the graph K2,5 � K1 is demonstrated in the fol-
lowing Figure 6.13.
u1'
u2
u2'
u1
3 5
119
7
12
1 2
8
64
10
13 14
v2' v3' v4' v5'v1'
v1 v2 v3 v4 v5
Figure 6.13: SDC labeling in K2,5 �K1
135
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Theorem 6.2.3. K3,n �K1 is sum divisor cordial graph.
Proof. Let V (K3,n � K1) = {uj, u′j | 1 ≤ j ≤ 3}⋃{vj, v′j | 1 ≤ j ≤ n}, where
V (K3,n) = {u1, u2, u3, vj | 1 ≤ j ≤ n} and u′1, u′2, u′3, v′j(1 ≤ j ≤ n) are the lately
inserted vertices to construct the graph K3,n �K1.
Let E(K3,n �K1) = {uju′j | 1 ≤ j ≤ 3}⋃{vjv′j | 1 ≤ j ≤ n}⋃{u1vj, u2vj, u3vj | 1 ≤j ≤ n}.Also it is to be noted that, |V (K3,n �K1)| = 2n+ 6 and |E(K3,n �K1)| = 4n+ 3.
Consider a bijection f : V (K3,n �K1)→ {1, 2, 3, . . . , 2n+ 6} defined as below.
f(uj) = j 1 ≤ j ≤ 3.
f(u′1) = 5.
f(u′2) = 4.
f(u′3) = 6.
f(vj) = 6 + (2j − 1). 1 ≤ j ≤ n;
f(v′j) = 6 + (2j); 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌈4n+ 3
2
⌉, ef (0) =
⌊4n+ 3
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, K3,n �K1 is SDC graph.
Example 6.2.3. SDC labeling in the graph K3,7 � K1 is demonstrated in the fol-
lowing Figure 6.14.
u1' u2' u3'64
8 12 14
1 3
5
1197 13
2
1715 19
1816 2010
v2' v3' v4' v5' v6'v1' v7'
v1 v2 v3 v6 v7v4 v5
u1 u2 u3
Figure 6.14: SDC labeling in K3,7 �K1
136
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Theorem 6.2.4. Wn �K1 is sum divisor cordial graph.
Proof. Let V (Wn � K1) = {vj | 0 ≤ j ≤ n}⋃{v′j | 0 ≤ j ≤ n}, where v0 is apex
vertex and vj(1 ≤ j ≤ n) are rim vertices of Wn.
Let v′j(0 ≤ j ≤ n) be the lately inserted vertices to construct the graph Wn �K1.
Let E(Wn �K1) = {v0vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{vjv′j |
0 ≤ j ≤ n}.Also it is to be noted that, |V (Wn �K1)| = 2n+ 2 and |E(Wn �K1)| = 3n+ 1.
Consider a bijection f : V (Wn �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.
Case 1: n ≡ 0, 2(mod 4).
f(v0) = 1.
f(v′0) = 2n+ 2.
f(v2j−1) = 4j − 2; 1 ≤ j ≤ n
2.
f(v2j) = 4j − 1; 1 ≤ j ≤ n
2.
f(v′2j−1) = 4j; 1 ≤ j ≤ n
2.
f(v′2j) = 4j + 1; 1 ≤ j ≤ n
2.
Case 2: n ≡ 1, 3(mod 4).
f(v0) = 1.
f(v′0) = 2n+ 1.
f(v2j−1) = 4j − 2; 1 ≤ j ≤ n+ 1
2.
f(v2j) = 4j − 1; 1 ≤ j ≤ n− 1
2.
f(v′2j−1) = 4j; 1 ≤ j ≤ n+ 1
2.
f(v′2j) = 4j + 1; 1 ≤ j ≤ n− 1
2.
By looking into the above prescribed pattern, the below table describes edge label
conditions.
137
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) =⌊3n+1
2
⌋, ef (0) =
⌈3n+1
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 3n+12
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Wn �K1 is SDC graph.
Example 6.2.4. SDC labeling in the graph W7�K1 is demonstrated in the following
Figure 6.15.
v0
v1
v7
v6
v5
v4
v7'
v1'
v6'
v5'
v4'
v3
v2
v3'
v2'
v0'
1
54
32
9
8
7
6
13
1211
10
15
16
14
Figure 6.15: SDC labeling in W7 �K1
Theorem 6.2.5. Hn �K1 is sum divisor cordial graph.
Proof. Let V (Hn �K1) = {v0, vj, uj | 1 ≤ j ≤ n}⋃{v′0, v′j, u′j | 1 ≤ j ≤ n}, where
v0 is apex vertex, vj, uj are vertices of the helm Hn and d(vj) = 4(1 ≤ j ≤ n),
d(uj) = 1(1 ≤ j ≤ n).
Let v′0, v′1, v′2, . . . , v
′n, u
′1, u′2, . . . , u
′n be the lately inserted vertices to construct the
graph Hn �K1.
Let E(Hn � K1) = {v0vj, viuj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n}⋃{vnv1}⋃
{v0v′0, viv′j, uju′j | 1 ≤ j ≤ n}.Also it is to be noted that, |V (Hn �K1)| = 4n+ 2 and |E(Hn �K1)| = 5n+ 1.
Consider a bijection f : V (Hn �K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.
Case 1: n ≡ 0, 1(mod 4).
f(v0) = 1.
f(v′0) = 4n+ 2.
138
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
f(vj) =
j + 1 j ≡ 0, 1(mod 4).
j + 2 j ≡ 2(mod 4).
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′j) = (n+ 1) + 2j; 1 ≤ j ≤ n.
f(uj) = n+ 2j; 1 ≤ j ≤ n.
f(u′j) = (3n+ 1) + i; 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(vj) =
j j ≡ 0, 1(mod 4).
j + 1 j ≡ 2(mod 4).
j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′j) = (n+ 1) + 2j; 2 ≤ j ≤ n.
f(v′1) = n+ 2.
f(u1) = n;
f(uj) = n+ 2j; 2 ≤ j ≤ n.
f(u′j) = (3n+ 1) + j; 1 ≤ j ≤ n.
Case 3: n ≡ 3(mod 4).
f(vj) =
j + 1 j ≡ 0, 1(mod 4)
j + 2 j ≡ 2(mod 4)
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v0) = 4n+ 2.
f(v′0) = 1.
f(v′j) = (n+ 1) + 2j; 1 ≤ j ≤ n.
f(uj) = n+ 2j; 1 ≤ j ≤ n.
f(u′j) = (3n+ 1) + j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern, the below table describes edge label
139
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
conditions.
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) =⌊5n+1
2
⌋, ef (0) =
⌈5n+1
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 5n+12
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Hn �K1 is SDC graph.
Example 6.2.5. SDC labeling in the graph H7�K1 is demonstrated in the following
Figure 6.16.
u1u2
u2'
v0
v1
v'1
v7
v7'
v6
v6'v5
v5'
v4v4'
v3
v3'v2
v'2
u7
u7'
u1'
u6
u5
u5'
u4 u6'
u4'
u3u3'
v0'1
3
24
5
7
6
8
910
21
22
20
19
17
13
18
15
12
16
14
11
23
27
29
28
26
25
24
30
Figure 6.16: SDC labeling in H7 �K1
Theorem 6.2.6. Fln �K1 is sum divisor cordial graph.
Proof. Let V (Fln �K1) = {v0, vj, uj, | 1 ≤ j ≤ n}⋃{v′0, v′j, u′j, | 1 ≤ j ≤ n}, where
v0 is apex vertex, vj, uj are the vertices in the flower graph Fln and d(vj) = 4(1 ≤j ≤ n), d(uj) = 2(1 ≤ j ≤ n) .
Let v′0, v′1, v′2, . . . , v
′n, u
′1, u′2, . . . , u
′n be the lately inserted vertices to construct the
graph Fln �K1.
Let E(Fln � K1) = {v0vj, v0uj, viuj | 1 ≤ j ≤ n.}⋃{vjvj+1 | 1 ≤ j ≤ n −1}⋃{vnv1}
⋃ {vjv′j, uju′j | 1 ≤ j ≤ n}⋃{v0v′0}.Also it is to be noted that, |V (Fln �K1)| = 4n+ 2 and |E(Fln �K1)| = 6n+ 1.
140
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Consider a bijection f : V (Fln �K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.
f(vj) =
j + 1 j ≡ 0, 1(mod 4).
j + 2 j ≡ 2(mod 4).
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v0) = 1.
f(v′0) = 4n+ 2.
Case 1: n ≡ 0(mod 4).
f(v′4j−3) = 3n+ 4j − 1; 1 ≤ j ≤ n
4.
f(v′4j−2) = 3n+ 4j + 1; 1 ≤ j ≤ n
4.
f(v′4j−1) = 3n+ 4j − 2; 1 ≤ j ≤ n
4.
f(v′4j) = 3n+ 4j; 1 ≤ j ≤ n
4.
f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n
2.
f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n
2.
f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n
2.
f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n
2.
Case 2: n ≡ 2(mod 4).
f(v′4j−3) = 3n+ 4j − 3; 1 ≤ j ≤ n+ 2
4.
f(v′4j−2) = 3n+ 4j − 1; 1 ≤ j ≤ n+ 2
4.
f(v′4j−1) = 3n+ 4j; 1 ≤ j ≤ n− 2
4.
f(v′4j) = 3n+ 4j + 2; 1 ≤ j ≤ n− 2
4.
f(u2j−1) = n+ 4j − 3; 1 ≤ j ≤ n
2.
f(u2j) = n+ 4j; 1 ≤ j ≤ n
2.
f(u′2j−1) = n+ 4j − 1; 1 ≤ j ≤ n
2.
f(u′2j) = n+ 4j + 2; 1 ≤ j ≤ n
2.
141
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Case 3: n ≡ 1(mod 4).
f(v′4j−3) = 3n+ 4j; 1 ≤ j ≤ n− 1
4.
f(v′4j−2) = 3n+ 4j + 2; 1 ≤ j ≤ n− 1
4.
f(v′4j−1) = 3n+ 4j − 3; 1 ≤ j ≤ n− 1
4.
f(v′4j) = 3n+ 4j − 1; 1 ≤ j ≤ n− 1
4.
f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n+ 1
2.
f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n+ 1
2.
f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n− 1
2.
f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n− 1
2.
f(v′n) = 4n.
Case 4: n ≡ 3(mod 4).
f(v′4j−3) = 3n+ 4j; 1 ≤ j ≤ n+ 1
4.
f(v′4j−2) = 3n+ 4j + 2; 1 ≤ j ≤ n− 3
4.
f(v′4j−1) = 3n+ 4j − 3; 1 ≤ j ≤ n− 3
4.
f(v′4j) = 3n+ 4j − 1; 1 ≤ j ≤ n− 3
4.
f(u2j−1) = n+ 4j − 2; 1 ≤ j ≤ n+ 1
2.
f(u′2j−1) = n+ 4j; 1 ≤ j ≤ n+ 1
2.
f(u2j) = n+ 4j − 1; 1 ≤ j ≤ n− 1
2.
f(u′2j) = n+ 4j + 1; 1 ≤ j ≤ n− 1
2.
f(v′n−1) = 4n− 2.
f(v′n) = 4n.
142
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
By looking into the above prescribed pattern,
ef (1) =
⌈6n+ 1
2
⌉, ef (0) =
⌊6n+ 1
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Fln �K1 is SDC graph.
Example 6.2.6. SDC labeling in the graph Fl7�K1 is demonstrated in the following
Figure 6.17.
v0
v1
v1'
v7
v7'
v6
v6'v5
v5'
v4v4'
v3
v3'v2
v2'
u1'
u1
u7u7'
u6
u6'
u5
u5'
u4
u4'
u3' u3
u2
u2'
v0'5
7
23
21
17
1
8
6
11
4
18
2
14
20
16
13
19
15 3
912
28
10
29
26
2527
22
24
30
Figure 6.17: SDC labeling in Fl7 �K1
Theorem 6.2.7. Fn �K1 is SDC graph.
Proof. Let V (Fn � K1) = {vj, v′j | 1 ≤ j ≤ n}⋃{u1, u′1}, where u1 is apex vertex,
vj(1 ≤ j ≤ n) are the vertices of the path Pn corresponding to the fan graph Fn and
let u′1, v′1, v′2, . . . , v
′n are lately inserted vertices to construct the graph Fn �K1.
Let E(Fn �K1) = {u1vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{viv′j | 1 ≤ j ≤n}⋃{u1u′1}.Also it is to be noted that, |V (Fn �K1)| = 2n+ 2 and |E(Fn �K1)| = 3n.
Consider a bijection f : V (Fn �K1)→ {1, 2, 3, . . . , 2n+ 2} defined as below.
143
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Case 1: n ≡ 0, 3(mod 4).
f(vj) =
j + 1 j ≡ 0, 1(mod 4).
j + 2 j ≡ 2(mod 4).
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(u1) = 1.
f(u′1) = 2n+ 2.
f(v′j) = (n+ 1) + j; 1 ≤ j ≤ n.
Case 2: n ≡ 1(mod 4).
f(vj) =
j + 1 j ≡ 0, 1(mod 4);
j + 2 j ≡ 2(mod 4);
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(u1) = 1.
f(u′1) = 2n+ 1.
f(v′j) = (n+ 1) + j; 1 ≤ j ≤ n− 1.
f(v′n) = 2n+ 2.
Case 3: n ≡ 2(mod 4).
f(vj) =
j + 1 j ≡ 0, 1(mod 4);
j + 2 j ≡ 2(mod 4);
j j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(u1) = 1.
f(u′1) = 2n+ 1.
f(v′j) = n+ 1 + j; 2 ≤ j ≤ n− 1.
f(v′1) = n+ 1.
f(v′n) = 2n+ 2.
144
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 1(mod 4) ef (1) =⌈3n2
⌋, ef (0) =
⌊3n2
⌉
n ≡ 3(mod 4) ef (1) =⌊3n2
⌋, ef (0) =
⌈3n2
⌉
n ≡ 0, 2(mod 4) ef (1) = 3n2
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Fn �K1 is SDC graph.
Example 6.2.7. SDC labeling in the graph F8�K1 is demonstrated in the following
Figure 6.18.
u1'
u1
64 8
1210 14
1
3 5
11
97
13
2
1715
18
16
v2' v3' v4' v5' v6'v1' v7'
v1 v2 v3 v6 v7v4 v5
v8'
v8
Figure 6.18: SDC labeling in F8 �K1
Theorem 6.2.8. DFn �K1 is SDC graph.
Proof. Let V (DFn � K1) = {vj, v′j, | 1 ≤ j ≤ n}⋃{uj, u′j | 1 ≤ j ≤ 2}, where
u1, u2 are apex vertices of degree n and vj(1 ≤ j ≤ n) are the vertices of path Pn
corresponding to the double fan DFn.
Let u′1, u′2, v′1, v′2, . . . , v
′n be the lately inserted vertices to construct the graph DFn�
K1.
Let E(DFn � K1) = {u1vj, u2vj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vjv′j |1 ≤ j ≤ n.}⋃{u1u′1}
⋃{u2u′2}.Also it is to be noted that, |V (DFn �K1)| = 2n+ 4 and |E(DFn �K1)| = 4n+ 1.
Consider a bijection f : V (DFn �K1)→ {1, 2, 3, . . . , 2n+ 4} defined as below.
145
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Case 1: n ≡ 0, 2(mod 4).
f(u1) = 1.
f(u′1) = 2n+ 4.
f(u2) = 2.
f(u′2) = 2n+ 3.
f(vj) = 2 + j; 1 ≤ j ≤ n.
f(v′j) = n+ 2 + j; 1 ≤ j ≤ n.
Case 2: n ≡ 1, 3(mod 4).
f(u1) = 1.
f(u′1) = 2n+ 2.
f(u2) = 2.
f(u′2) = 2n+ 4.
f(vj) = 2 + j; 1 ≤ j ≤ n.
f(v′2j−1) = n+ (2j − 1) + 3; 1 ≤ j ≤ n+ 1
2.
f(v′2j) = n+ 2j + 1; 1 ≤ j ≤ n− 1
2.
By looking into the above prescribed pattern, the below table describes edge label
conditions.
Cases of n Edge label conditions
n ≡ 1, 3(mod 4) ef (1) =⌈4n+1
2
⌉, ef (0) =
⌊4n+1
2
⌋
n ≡ 0, 2(mod 4) ef (1) =⌊4n+1
2
⌋, ef (0) =
⌈4n+1
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, DFn �K1 is SDC graph.
Example 6.2.8. SDC labeling in the graph DF6 � K1 is demonstrated in the fol-
lowing Figure 6.19.
146
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
u2
u2'
3 4 5 6 7 8
1
2
16
15
9 10 11 12 13 14v2' v3' v4' v5' v6'v1'
v1 v2 v3 v6v4 v5
u1
u1'
Figure 6.19: SDC labeling in DF6 �K1
Theorem 6.2.9. S(K1,n)�Kn is SDC graph.
Proof. Let V (S(K1,n) � K1) = {v0, v′0}⋃{vj, uj, v′j, u′j | 1 ≤ j ≤ n}, where v0 is
apex vertex, vj, uj are the vertices of the graph S(K1,n) and d(vj) = 2(1 ≤ j ≤ n),
d(uj) = 1(1 ≤ j ≤ n) .
Let v′0, v′j, u′j (1 ≤ i ≤ n) be the lately inserted vertices to construct the graph
S(K1,n) �K1.
Let E(S(K1,n)�K1) = {v0vj, vjuj | 1 ≤ j ≤ n.}⋃{v0v′0, viv′j, uju′j | 1 ≤ j ≤ n.}Also it is to be noted that, |V (S(K1,n)�K1)| = 4n+2 and |E(S(K1,n)�K1)| = 4n+1.
Consider a bijection f : V (S(K1,n)�K1)→ {1, 2, 3, . . . , 4n+ 2} defined as below.
f(v0) = 1.
f(v′0) = 4n+ 2.
f(vj) = 2j + 1; 1 ≤ j ≤ n.
f(v′j) = 2j; 1 ≤ j ≤ n.
f(uj) = (2n+ 1) + 2j; 1 ≤ j ≤ n.
f(u′j) = 2n+ 2j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌊4n+ 1
2
⌋, ef (0) =
⌈4n+ 1
2
⌉.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, S(K1,n)�K1 is SDC graph.
147
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
Example 6.2.9. SDC labeling in the graph S(K1,5) � K1,5 is demonstrated in the
following Figure 6.20.
u1
v0 v1 v1'
v5
v5'
v4
v4'
v3
v3'v2
v2'
u5
u5'
u1'
u4
u3
u3'
u2
u4'
u2'
v0'
1
3 2
10
9
8
6
7
4
5
12
20
13
2111
18
19
16
17
14
22
15
Figure 6.20: SDC labeling in S(K1,5)�K1,5
Theorem 6.2.10. G�K1 is SDC graph, where G is cycle with one chord.
Proof. Let V (G�K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of Cn
and v′j(1 ≤ j ≤ n) are the lately inserted vertices to construct the graph G�K1.
Let E(G �K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤ n}⋃{v2vn},
where v2vn is the chord of Cn.
Also it is to be noted that, |V (G�K1)| = 2n and |E(G�K1)| = 2n+ 1.
Consider a bijection f : V (G�K1)→ {1, 2, 3, . . . , 2n} defined as below.
f(vj) = 2j − 1; 1 ≤ j ≤ n.
f(v′j) = 2j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern
ef (1) =
⌈2n+ 1
2
⌉, ef (0) =
⌊2n+ 1
2
⌋.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, G�K1 is SDC graph, where G is cycle with one chord.
Example 6.2.10. SDC labeling in corona of C6 with one chord and K1 is demon-
strated in the following Figure 6.21.
148
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
v1
v1'
v2
v2'
v3
v3' v4
v4'
v5
v5'
v6
v6'1
2
3
4
5
67
8
10
9
1112
Figure 6.21: SDC labeling in corona of C6 with one chord and K1
Theorem 6.2.11. Cn,3 �K1 is SDC graph.
Proof. Let V (Cn,3 �K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of
Cn and v′j(1 ≤ j ≤ n) are lately inserted vertices to construct the graph Cn,3 �K1.
Let E(Cn,3 � K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤
n}⋃{v2vn, v2vn−1}, where v2vn and v2vn−1 are chords of Cn.
Also it is to be noted that, |V (Cn,3 �K1)| = 2n and |E(Cn,3 �K1)| = 2n+ 2.
Consider a bijection f : V (Cn,3 �K1)→ {1, 2, 3, . . . , 2n} define defined as below.
Case 1: n ≡ 0, 1, 3(mod 4).
f(vj) =
j j ≡ 0, 1(mod 4);
j + 1 j ≡ 2(mod 4);
j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′j) = n+ j; 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(vj) =
j j ≡ 0, 1(mod 4);
j + 1 j ≡ 2(mod 4);
j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′1) = n.
f(v′2) = n+ 2.
f(v′j) = n+ j; 3 ≤ j ≤ n.
149
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
By looking into the above prescribed pattern,
ef (1) = n+ 1 = ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn,3 �K1 is SDC graph.
Example 6.2.11. SDC labeling in the graph C7,3 �K1 is demonstrated in the fol-
lowing Figure 6.22.
v1
v1'
v2
v2'
v3v3'
v4
v4' v5
v5'
v6 v6'
v7
v7'1
8
3
9
210
4
11
5
12
7 13
6
14
Figure 6.22: SDC labeling in C7,3 �K1
Theorem 6.2.12. Cn(1, 1, n− 5)�K1 is SDC graph.
Proof. Let V (Cn(1, 1, n− 5)�K1) = {vj, v′j | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are
the vertices of Cn and v′j(1 ≤ j ≤ n) are lately inserted vertices to construct the
graph Cn(1, 1, n− 5)�K1.
Let E(Cn(1, 1, n − 5) �K1) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{vjv′j | 1 ≤ j ≤
n}⋃{v1v3}⋃{v3vn−1}
⋃{vn−1v1}, where u1u3, u3un−1 and u1un−1 are chords of Cn
which by themselves form a triangle.
Also it is to be noted that, |V (Cn(1, 1, n− 5)�K1)| = 2n and |E(Cn(1, 1, n− 5)�K1)| = 2n+ 3.
Consider a bijection f : V (Cn(1, 1, n−5)�K1)→ {1, 2, 3, . . . , 2n} defined as below.
Case 1: n ≡ 0, 1, 3(mod 4).
f(v′j) = n+ j; 1 ≤ j ≤ n.s
150
6.2. SDC Labeling in the Graphs constructed from Corona Product with K1
f(vj) =
j j ≡ 0, 1(mod 4);
j + 1 j ≡ 2(mod 4);
j − 1 j ≡ 3(mod 4); 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(v′1) = n.
f(v′2) = n+ 2.
f(vj) =
j i ≡ 0, 1(mod 4);
j + 1 i ≡ 2(mod 4);
j − 1 i ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′j) = n+ j; 3 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) =
⌊2n+ 3
2
⌋, ef (0) =
⌈2n+ 3
2
⌉.
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, Cn(1, 1, n− 5)�K1 is SDC graph.
Example 6.2.12. SDC labeling in the graph C8(1, 1, 3)�K1 is demonstrated in the
following Figure 6.23.
v1
v1'
v2
v2'
v3v3'
v4
v4'
v5
v5'
v6
v6'
v8
v7'v7
v8'1
3
2
45
7
6
8
9
10
11
12
13
16
15
14
Figure 6.23: SDC labeling in C8(1, 1, 3)�K1
151
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
6.3 SDC Labeling With the Use of Switching of a Vertex in
Cycle Allied Graphs
Vaidya and Shah[49] derived some captivating results on divisor cordial labeling
of the graphs constructed from switching a vertex in different graphs. In current
segment we demonstrate some graphs constructed from switching invariance in cycle
allied graphs for SDC labeling.
Theorem 6.3.1. Gv is SDC, where G is cycle Cn with one chord.
Proof. Let cycle Cn with one chord be denoted as G.
Let V (G) = {vj | 1 ≤ j ≤ n}, where vj(1 ≤ j ≤ n) are vertices of Cn.
Let E(G) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}, where v2vn is the chord of
Cn. WLOG let the switched vertex be v1 (of degree 2 or 3).
Let Gv1 denote the graph constructed from switching of vertex v1.
Corresponding to the vertices of different degree in Cn with one chord, it is required
to discuss following two cases. Case 1: d(v1) = 2.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v1vj | 3 ≤ j ≤ n− 1}.
It is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 4.
Consider a bijection f from V (Gv1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 0(mod 4).
f(v1) = 1.
f(v2) = 2.
f(vj) =
j ; j ≡ 3(mod 4)
j + 1 ; j ≡ 1, 0(mod 4)
j + 2 ; j ≡ 2(mod 4); 3 ≤ j ≤ n− 1.
f(vn) = 4.
152
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Subcase 2: n ≡ 1(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 3: n ≡ 2(mod 4).
f(v1) = n.
f(v2) = 2.
f(vj) =
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4)
j ; j ≡ 1, 2(mod 4); 3 ≤ j ≤ n− 1.
f(vn) = 1.
Subcase 4: n ≡ 3(mod 4).
f(v1) = 1.
f(v2) = 2.
f(vj) =
j ; j ≡ 3(mod 4)
j + 1 ; j ≡ 0, 1(mod 4)
j + 2 ; j ≡ 2(mod 4); 3 ≤ j ≤ n− 2.
f(vn−1) = n.
f(vn) = 4.
By looking into the above prescribed pattern,
ef (1) = ef (0) = n− 2.
Case 2: d(v1) = 3.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 4 ≤ j ≤ n− 1}.
153
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Also it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.
Consider a bijection f from V (Gv1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 0, 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 3(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n.
By looking into the above prescribed pattern, ef (1) = ef (0) = n− 3.
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, Gv is SDC, where G is cycle Cn with one chord.
Example 6.3.1. The following Figure 6.24 demonstrates
(i) Cycle C7 with one chord.
(ii) SDC labeling in (G)v, where d(v) = 2 and G is C7 with one chord.
(iii) SDC labeling in (G)v, where d(v) = 3 and G is C7 with one chord.
1
2
6
7
4
3
5
v6
v5v4
v3
v2
v1
v7
1
3
4
7
2
5 6
Figure 6.24: SDC labeling in (G)v.
154
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Theorem 6.3.2. (Cn,3)v is SDC.
Proof. Let V (Cn,3) = {v1, v2, . . . , vn}, where vj(1 ≤ j ≤ n) are the vertices of Cn.
Let E(Cn,3) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v2vn}
⋃{v2vn−1}, where v2vn,
v2vn−1 are chords.
WLOG let v1 be the switched vertex.
Let (Cn,3)v1 denote the graph constructed from switching of vertex v1 of Cn,3.
Corresponding to the vertices of different degree in Cn,3, it is required to discuss
following three cases.
Case 1: d(v1) = 2.
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2vn}⋃{v2vn−1}
⋃{v1vj | 3 ≤ j ≤ n− 1}.In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 3.
Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 0, 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 3(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 1, 3(mod 4) ef (1) = n− 2, ef (0) = n− 1
n ≡ 2(mod 4) ef (1) = n− 1, ef (0) = n− 2
Case 2: d(v1) = 3.
155
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn}⋃{v1vj | 4 ≤ j ≤ n− 1}.
In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 5.
Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 0, 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 3(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2, 3(mod 4) ef (1) = b2n−52c, ef (0) = d2n−5
2e
n ≡ 1(mod 4) ef (0) = b2n−52c, ef (1) = d2n−5
2e
Case 3: d(v1) = 4.
Then by the effect of switching operation, the edge set of (Cn,3)v1 is
E((Cn,3)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v1vj | 3 ≤ j ≤ n− 3}.In this case it is to be noted that, |V ((Cn,3)v1)| = n and |E((Cn,3)v1)| = 2n− 7.
Consider a bijection f from V ((Cn,3)v1) to {1, 2, 3, . . . , n} define defined as below.
Subcase 1: n ≡ 0, 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
156
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Subcase 2: n ≡ 3(mod 4).
f(vn) = n.
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 1(mod 4) ef (0) =⌊2n−7
2
⌋, ef (1) =
⌈2n−7
2
⌉
n ≡ 2, 3(mod 4) ef (1) =⌊2n−7
2
⌋, ef (0) =
⌈2n−7
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Cn,3)v is SDC graph.
Example 6.3.2. The following Figure 6.25 demonstrates
(i) The graph C8,3.
(ii) SDC labeling in (C8,3)v, where d(v) = 2.
(iii) SDC labeling in (C8,3)v, where d(v) = 3.
(iv) SDC labeling in (C8,3)v, where d(v) = 4.
1
5
72
4
3 6
8
1
5
7
2
4
3
6
8
v7
v6
v5
v4
v3
v1
v8v2 1
3
7
2
4 5
6
8
Figure 6.25: SDC labeling in (C8,3)v.
Theorem 6.3.3. (Cn(1, 1, n− 5))v is SDC (n ≥ 6, n ∈ N) .
Proof. Let Cn(1, 1, n− 5) be denoted as G.
Let V (G) = {vj | 1 ≤ j ≤ n} = V (Cn).
157
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Let E(G) = {vjvj+1 | 1 ≤ j ≤ n−1}⋃{vnv1}⋃{v1v3}
⋃{v3vn−1}⋃{vn−1v1}, where
v1vn−1, v1v3, vn−1v3 are chords.
WLOG let v1 be the switched vertex.
Let Gv1 denote the graph constructed from switching of arbitrary vertex v1 of G.
As per different possible degrees of vertices in the graph G, we need to consider
following two cases.
Corresponding to the vertices of different degree in Cn(1, 1, n− 5), it is required to
discuss following two cases.
Case 1: d(v1) = 2.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v2v4}⋃{v4vn−1}
⋃{vn−1v2}⋃{v1vj | 3 ≤ j ≤
n− 1}.In this case it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 2.
Consider a bijection f from V (Gv1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 3(mod 4).
f(v1) = n− 1.
f(vj) =
j − 1 ; j ≡ 1, 2(mod 4)
j ; j ≡ 3(mod 4)
j − 2 ; j ≡ 0(mod 4); 2 ≤ j ≤ n.
By looking into the above prescribed pattern, ef (1) = ef (0) = n− 1.
Case 2: d(v1) = 4.
Then by the effect of switching operation, the edge set of Gv1 is
E(Gv1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v3vn−1}⋃{v1vj | 4 ≤ j ≤ n− 2}.
In this case it is to be noted that, |V (Gv1)| = n and |E(Gv1)| = 2n− 6.
158
6.3. SDC Labeling With the Use of Switching of a Vertex in Cycle Allied Graphs
Consider a bijection f from V (Gv1) to {1, 2, 3, . . . , n} defined as below.
Subcase 1: n ≡ 1, 2(mod 4).
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 3(mod 4).
f(v1) = n− 1.
f(vj) =
j ; j ≡ 1, 2(mod 4)
j + 1 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 2 ≤ j ≤ n− 2.
f(vn−1) = n.
f(vn) = 1.
By looking into the above prescribed pattern, ef (1) = ef (0) = n− 3.
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Cn(1, 1, n− 5))v is SDC.
Example 6.3.3. The following Figure 6.26 demonstrates
(i) The graph C7(1, 1, 2).
(ii) SDC labeling in (C7(1, 1, 2))v, where d(v) = 2.
(iii) SDC labeling in (C7(1, 1, 2))v, where d(v) = 4.
v6
v5v4
v3
v2
v1
v7 1
7
6
53
4
2
1
4
7
6
5
2
3
Figure 6.26: SDC labeling in (C7(1, 1, 2))v
159
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
6.4 SDC Labeling With the Use of Switching of a Vertex in
Wheel and Shell Allied Graphs
In the current segment we demonstrate some new graphs constructed from switching
of a vertex in wheel and shell allied graphs for SDC labeling.
Theorem 6.4.1. (Wn)v is SDC, where v is rim vertex.
Proof. Let V (Wn) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)
are the rim vertices of wheel Wn.
Let E(Wn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}.
Let (Wn)v1 denote the graph constructed from switching of a rim vertex v1 of Wn.
Then by the effect of switching operation, the edge set of (Wn)v1 is
E((Wn)v1) = {vjvj+1 | 2 ≤ j ≤ n− 1}⋃{v0vj | 2 ≤ j ≤ n}⋃{v1vj | 3 ≤ j ≤ n− 1}.In this case it is to be noted that, |V ((Wn)v1)| = n+ 1 and |E((Wn)v1)| = 3n− 6.
Consider a bijection f from V ((Wn)v1) to {1, 2, 3, . . . , n+ 1} defined as below.
Case 1: n ≡ 0, 1(mod 4).
f(v0) = 1.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(v0) = 1.
f(v1) = 3.
f(vj) =
j ; j ≡ 2(mod 4)
j + 1 ; j ≡ 0, 3(mod 4)
j + 2 ; j ≡ 1(mod 4); 2 ≤ j ≤ n.
160
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Case 3: n ≡ 3(mod 4).
f(v0) = 1.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n+ 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) = 3n−62
= ef (1)
n ≡ 1(mod 4) ef (0) =⌊3n−6
2
⌋, ef (1) =
⌈3n−6
2
⌉
n ≡ 3(mod 4) ef (0) =⌈3n−6
2
⌉, ef (1) =
⌊3n−6
2
⌋
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, (Wn)v is SDC, where v is rim vertex.
Example 6.4.1. The following Figure 6.27 demonstrates
(i) Wheel W9.
(ii) SDC labeling in (W9)v1, where v1 is rim vertex.
v1
v9
v8
v7
v6v5
v4
v3
v2
v0
9
7
5
3
1
2
4
6
8
10
Figure 6.27: SDC labeling in (W9)v1 .
Remark 6.4.1. Switching apex vertex in Wn, the resultant graph is Cn⋃K1 which
is SDC graph!!
161
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Theorem 6.4.2. (Gn)v is SDC, where v is not apex vertex.
Proof. Let V (Gn) = {vj | 0 ≤ j ≤ 2n}, where v0 is apex vertex and vj(1 ≤ j ≤ 2n)
are other vertices of Gn, where
deg(vj) =
2 when j is even;
3 when j is odd; 1 ≤ j ≤ 2n.
Let E(Gn) = {vjvj+1 | 1 ≤ j ≤ 2n− 1}⋃{v2nv1}⋃{v0v2j−1 | 1 ≤ j ≤ n}.
(Gn)vi∼= (Gn)vj , where d(vi) = d(vj).
Let (Gn)vj denote the graph constructed from switching of vertex vj (j = 1, 2) of
Gn.
Corresponding to the vertices of different degree in Gn, it is required to discuss
following two cases.
Case 1: deg(v1) = 3.
Then by the effect of switching operation, the edge set of (Gn)v1 is
E((Gn)v1) = {v0v2j−1 | 2 ≤ j ≤ n}⋃{vjvj+1 | 2 ≤ j ≤ 2n − 1}⋃{v1vj | 3 ≤ j ≤2n− 1}.In this case it is to be noted that, |V (Gn)v1)| = 2n+ 1 and |E(Gn))v1)| = 5n− 6.
Consider a bijection f from V ((Gn))v1)) to {1, 2, . . . , 2n+ 1} defined as below.
f(v0) = 1.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) = 5n−62
= ef (0)
n ≡ 1, 3(mod 4) ef (0) =⌊5n−6
2
⌋, ef (1) =
⌈5n−6
2
⌉
Case 2: deg(v2) = 2.
Then by the effect of switching operation, the edge set of (Gn)v2 is
E((Gn)v2) = {v0v2j−1 | 1 ≤ j ≤ n}⋃{vjvj+1 | 3 ≤ j ≤ 2n − 1}⋃{v2nv1}⋃{v2vj |
162
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
4 ≤ j ≤ 2n}.In this case it is to be noted that, |V ((Gn)v2)| = 2n+ 1 and |E((Gn)v2)| = 5n− 4.
Consider a bijection f from V ((Gn)v2) to {1, 2, . . . , 2n+ 1} defined as below.
Subcase 1: n ≡ 0, 2(mod 4).
f(v0) = 1.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
Subcase 2: n ≡ 1, 3(mod 4).
f(v0) = 1.
f(v1) = 3.
f(vj) =
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 2(mod 4)
j + 1 ; j ≡ 0, 1(mod 4); 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) =⌊5n−4
2
⌋, ef (0) =
⌈5n−4
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 5n−42
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Gn)v is SDC, v is not apex vertex.
Example 6.4.2. The following Figure 6.28 demonstrates
(i) Gear graph G6.
(ii) SDC labeling in (G6)v, where d(v) = 3.
(iii) SDC labeling in (G6)v, where d(v) = 2.
163
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
v11
v0
v12
v10
v9
v8
v7
v6
v5
v4
v2
v1
v3 13
7
1
5
9
2
3
4
68
10
11
12 2
1
3
5
46
79
8
10
11
1312
Figure 6.28: SDC labeling in (G6)v.
Remark 6.4.2. Switching apex vertex in Gn, the resultant graph is C2n
⋃K1 which
is SDC graph!!
Theorem 6.4.3. (Sn)v is SDC, where v is not apex vertex.
Proof. Let V (Sn) = {vi | 1 ≤ i ≤ n}, where v1 is apex vertex and vj(2 ≤ j ≤ n) are
the other vertices of shell Sn, where
deg(vj) =
2 when j = 2, n.
3 when ; 3 ≤ j ≤ n− 1.
Let E(Sn) = {vjvj+1 | 1 ≤ j ≤ n− 1}⋃{vnv1}⋃{v1vj | 3 ≤ j ≤ n− 1}.
(Sn)vi∼= (Sn)vj , where d(vi) = d(vj).
Let (Sn)vj denote the graph constructed from switching of vertex vj (j = 2, 3) of Sn.
Corresponding to the vertices of different degree in Sn, it is required to discuss
following two cases.
Case 1: deg(v3) = 3.
Then by the effect of switching operation, the edge set of (Sn)v3 is
Let E((Sn)v3) = {v1vj | 4 ≤ j ≤ n− 1}⋃{vjvj+1 | 4 ≤ j ≤ n− 1}⋃{vnv1}{v1v2}⋃
{v3vj | 5 ≤ j ≤ n}.In this case it is to be noted that, |V ((Sn)v3)| = n and |E((Sn)v3)| = 3n− 10.
Consider a bijection f from V ((Sn)v3) to {1, 2, . . . n} defined as below.
Subcase 1: n ≡ 0(mod 4).
f(v1) = 1.
164
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
f(vn) = n− 1.
f(vn−1) = 2.
f(vj) =
j ; j ≡ 3(mod 4)
j + 1 ; j ≡ 0, 1(mod 4)
j + 2 ; j ≡ 2(mod 4); 2 ≤ j ≤ n− 2.
Subcase 2: n ≡ 1, 3(mod 4).
f(v1) = 1.
f(vn) = n.
f(vn−1) = 2.
f(vj) =
j ; j ≡ 3(mod 4)
j+ ; j ≡ 0, 1(mod 4)
j + 2 ; j ≡ 2(mod 4); 2 ≤ j ≤ n− 2.
Subcase 3: n ≡ 2(mod 4).
f(v1) = 1.
f(vn) = n.
f(vn−1) = 2.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 2 ≤ j ≤ n− 2.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) = 3n−102
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌊3n−10
2
⌋, ef (0) =
⌈3n−10
2
⌉
Case 2: deg(v2) = 2.
Then by the effect of switching operation, the edge set of (Sn)v2 is
165
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Let E((Sn)v2) = {v1vj | 3 ≤ j ≤ n − 1}⋃{vjvj+1 | 3 ≤ j ≤ n − 1}⋃{vnv1}{v2vj |4 ≤ j ≤ n}.In this case it is to be noted that, |V ((Sn)v2)| = n, |E((Sn)v2)| = 3n− 8.
Consider a bijection f from V ((Sn)v2) to {1, 2, . . . , n} defined as below.
Subcase 1: n ≡ 0, 1, 3(mod 4).
f(v1) = 1.
f(vn) = 2.
f(vj) =
j + 2 ; j + 2 ≡ 2(mod 4)
j ; j ≡ 3(mod 4)
j + 1 ; j ≡ 0, 1(mod 4); 3 ≤ j ≤ n− 1.
Subcase 2: n ≡ 2(mod 4).
f(v1) = 1.
f(v2) = 2.
f(vj) =
j + 1 ; j ≡ 1, 2(mod 4)
j + 2 ; j ≡ 3(mod 4)
j ; j ≡ 0(mod 4); 3 ≤ j ≤ n− 1.
f(vn) = n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0(mod 4) ef (0) =⌊3n−8
2
⌋, ef (1) =
⌈3n−8
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 3n−82
= ef (0)
n ≡ 2(mod 4) ef (1) =⌊3n−8
2
⌋, ef (0) =
⌈3n−8
2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Sn)v is SDC, v is not apex vertex.
Example 6.4.3. The following Figure 6.29 demonstrates
(i) Shell graph S7.
166
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
(ii) SDC labeling in (S7)v, where d(v) = 3.
(iii) SDC labeling in (S7)v, where d(v) = 2.
v6
v5v4
v3
v7
v1
v2
2
1
3
4
5
6
72
1
4 3
5
67
Figure 6.29: SDC labeling in (S7)v
Remark 6.4.3. Switching apex vertex in Sn, the resultant graph is Pn−2⋃K1 which
is SDC graph!!
Theorem 6.4.4. (Hn)v is SDC, v is apex vertex.
Proof. Let V (Hn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤ n)
are the vertices of corresponding to cycle Cn and uj(1 ≤ j ≤ n) are the pendant
vertices.
E(Hn) = {vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}⋃{v0vj | 1 ≤ j ≤ n}⋃{vjuj | 1 ≤ j ≤
n}.Let (Hn)v0 denote the graph constructed from switching of apex vertex v0 of Hn.
Then by the effect of switching operation, the edge set of (Hn)v0 is
Let E((Hn)v0) = {v0uj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}{vjuj | 1 ≤j ≤ n}.In this case it is to be noted that, |V ((Hn)v0)| = 2n+ 1, |E((Hn)v0)| = 3n.
Consider a bijection f from V ((Hn)v0) to {1, 2, . . . 2n+ 1} defined as below.
Case 1: n ≡ 0, 2(mod 4).
f(v0) = 2n+ 1.
f(vj) = j; 1 ≤ j ≤ n.
f(uk) = n+ 1 + k; 1 ≤ k ≤ n.
167
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Case 2: n ≡ 1, 3(mod 4).
f(v0) = 2n.
f(vj) =
2j − 1 ; j ≡ 1, 3(mod 4)
2j − 2 ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n.
f(uk) =
2k + 1 ; k ≡ 1, 3(mod 4)
2k ; k ≡ 0, 2(mod 4); 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) = 3n2
= ef (0)
n ≡ 1, 3(mod 4) ef (0) =⌊3n2
⌋, ef (1) =
⌈3n2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, (Hn)v is SDC, where v is apex vertex.
Example 6.4.4. The following Figure 6.30 demonstrates
(i) Helm graph H6.
(ii) SDC labeling in (H6)v, where v is apex vertex.
13v0
v5
v4
v3
v1
v6
u5
u4
u3
u2
u1
u6
v2
1
2
3
4
6
7
8
9
10
12
5
11
Figure 6.30: SDC labeling in (H6)v.
Theorem 6.4.5. (CHn)v is SDC, where v is apex vertex.
Proof. Let V (CHn) = {v0, vj, uj | 1 ≤ j ≤ n}, where v0 is apex vertex, vj(1 ≤ j ≤ n)
are the vertices of inner cycle and uj(1 ≤ j ≤ n) are the vertices of outer cycle of
168
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
CHn.
Let E(CHn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n−1}⋃{v0vj, vjuj | 1 ≤ j ≤ n}⋃{ujuj+1, unu1 |1 ≤ j ≤ n− 1}.Let (CHn)v0 denote the graph constructed from switching of apex vertex v0.
Then by the effect of switching operation, the edge set of (CHn)v0 is
Let E((CHn)v0) = {v0uj | 1 ≤ j ≤ n}⋃{vjvj+1 | 1 ≤ j ≤ n − 1}⋃{vnv1}{vjuj |1 ≤ j ≤ n}{ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}.In this case it is to be noted that, |V ((CHn)v0)| = 2n+ 1 and |E((CHn)v0)| = 4n.
Consider a bijection f from V ((CHn)v0) to {1, 2, . . . 2n+ 1} define defined as below.
f(v0) = 1.
f(vj) = 2j + 1; 1 ≤ j ≤ n.
f(uk) = 2k; 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n = ef (1).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, (CHn)v is SDC, where v is apex vertex.
Example 6.4.5. The following Figure 6.31 demonstrates
(i) Closed helm graph CH6.
(ii) SDC labeling in (CH6)v, where v is apex vertex.
v0
12
13
10
11
8
9 6
7
5
4
2
v5
v4
v3
v2
v1
v6
u5
u4
u3
u2
u1
u6
1
3
Figure 6.31: SDC labeling in (CH6)v
169
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Theorem 6.4.6. (Fln)v is SDC, where v is not apex vertex.
Proof. Let V (Fln) = {u0}⋃{uj | 1 ≤ j ≤ n}⋃{vj | 1 ≤ j ≤ n}, where u0 is apex
vertex, uj(1 ≤ j ≤ n) are the internal vertices and vj(1 ≤ j ≤ n) are the external
vertices. deg(uj) = 4; deg(vj) = 2, 1 ≤ j ≤ n.
Let E(Fln) = {ujuj+1 | 1 ≤ j ≤ n− 1}⋃{unu1}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 1 ≤
j ≤ n}⋃{ujvj | 1 ≤ j ≤ n}(Fln)ui
∼= (Fln)uj , where d(ui) = d(uj).
Let (Fln)u1 and (Fln)v1 denote the graph constructed from switching of vertex u1
and v1 of Fln respectively.
Corresponding to the vertices of different degree in Fln, it is required to discuss
following two cases.
Case 1: deg(v1) = 2.
Then by the effect of switching operation, the edge set of (Fln)v1 is
E((Fln)v1) = {ujuj+1 | 1 ≤ j ≤ n}⋃{u0uj | 1 ≤ j ≤ n}⋃{u0vj | 2 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{v1uj | 2 ≤ j ≤ n− 1}⋃{v1vj | 2 ≤ j ≤ n}.In this case it is to be noted that, |V ((Fln)v1)| = 2n+ 1 and |E((Fln)v1)| = 6n− 4.
Consider a bijection f from V ((Fln)v1) to {1, 2, . . . 2n+ 1} defined as below.
f(u0) = 2.
f(v1) = 1.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
f(vj) = 2j; 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 2 = ef (0).
Case 2: deg(u1) = 4.
Then by the effect of switching operation, the edge set of (Fln)u1 is
E((Fln)u1) = {ujuj+1, unu1 | 2 ≤ j ≤ n− 1}⋃{u0uj | 2 ≤ j ≤ n}⋃{u0vj | 1 ≤ j ≤n}⋃{ujvj | 2 ≤ j ≤ n}⋃{u1uj | 3 ≤ j ≤ n− 1}⋃{u1vj | 2 ≤ j ≤ n}.In this case it is to be noted that, |V ((Fln)u1)| = 2n+ 1, |E((Fln)u1)| = 6n− 8.
170
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Consider a bijection f from V ((Fln)u1) to {1, 2, . . . 2n+ 1} defined as below.
f(u0) = 2.
f(u1) = 1.
f(v1) = 2n+ 1.
f(uj) = 2j; 2 ≤ j ≤ n.
f(vj) = 2j − 1; 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 4 = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Fln)v is SDC, v is not apex vertex.
Example 6.4.6. The following Figure 6.32 demonstrates
(i) Flower graph Fl4.
(ii) SDC labeling in (Fl4)v1, where d(v1) = 2.
(iii) SDC labeling in (Fl4)u1, where d(u1) = 4.
1
3
8
2
9`
64
5 7
2
1
7
8
5`
64
9
3
v1v4
u1 u4
v2v3
u2 u3
u0
Figure 6.32: SDC labeling in (Fl4)v
Remark 6.4.4. Switching apex vertex in Fln, the resultant graph is (Cn�K1)⋃K1
which is SDC graph!!
171
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Theorem 6.4.7. (Bm,n)v is SDC.
Proof. Let V (Bm,n) = {u0, v0, ui, vj | 1 ≤ i ≤ m, 1 ≤ j ≤ n}, where u0, v0 are apex
vertices and d(ui) = 1(1 ≤ i ≤ m) and d(vj) = 1(1 ≤ j ≤ n).
E(Bm,n) = {u0ui, v0vj, u0v0 | 1 ≤ i ≤ m, 1 ≤ j ≤ n}.WLOG, let us assume that m ≤ n (as Bm,n and Bn,m are isomorphic graphs).
Let (Bm,n)v denote the graph constructed from switching of an arbitrary vertex v
in Bm,n.
According to different degrees of vertices in (Bm,n)v it is required to discuss following
three cases.
Case 1: deg(vj) = 1.
WLOG, let us assume that v1 is the switched pendant vertex.
In this case it is to be noted that, |V ((Bm,n)v1)| = m+n+2, |E((Bm,n)v1)| = 2m+2n.
Consider a bijection f from V ((Bm,n)v1) to {1, 2, . . .m+ n+ 2} defined as below.
f(u0) = 2.
f(u1) = 3.
f(v0) = 4.
f(v1) = 1.
f(ui) = 3 + i; 2 ≤ i ≤ m.
f(vj) = 2 +m+ j; 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = m+ n = ef (0).
Case 2: deg(u0) = m.
In this case it is to be noted that, |V ((Bm,n)u0)| = m+n+2 and |E((Bm,n)u0)| = 2n.
Consider a bijection f from V ((Bm,n)u0) to {1, 2, . . . ,m+ n+ 2} defined as below.
f(u0) = 2.
f(v0) = 1.
f(ui) = 2 + n+ i; 1 ≤ i ≤ m.
172
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
f(vj) = 2 + j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = n = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
Case 3: Switching of apex vertex v0 of degree n.
In this case it is to be noted that, |V ((Bm,n)v0)| = m+n+2 and |E((Bm,n)v0)| = 2m.
Consider a bijection f from V ((Bm,n)v0) to {1, 2, . . .m+ n+ 2} defined as below.
f(u0) = 2.
f(v0) = 1.
f(ui) = 2 + n+ i; 1 ≤ i ≤ m.
f(vj) = 2 + j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern, ef (1) = m = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, (Bm,n)v is SDC.
Example 6.4.7. Bistar B4,5 and SDC labeling in (B4,5)v1, where d(v1) = 1 are
demonstrated in the following Figure 6.33.
u0
v0
u2
u3
v1
u4
v2
v3
v4
v5
u1
2
4
3
5
6
7
1
8
9
10
11
Figure 6.33: SDC labeling in (B4,5)v1
Example 6.4.8. Bistar B4,5 and SDC labeling in (B4,5)u0, where u0 is apex vertex
are demonstrated in the following Figure 6.34.
173
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
u1
u2
v1
v2
u3
u4
v3
v4
v5
v0
u0
1
2
3
4
5
6
7
8
9
10
11
Figure 6.34: SDC labeling in (B4,5)u0
Example 6.4.9. Bistar B4,5 and SDC labeling in (B4,5)v0, where v0 is apex vertex
are demonstrated in the following Figure 6.35.
u1
u2
v1
v2
u3
u4
v3
v4
v5
v0
u0
1
2
3
4
5
6
7
8
9
10
11
Figure 6.35: SDC labeling in (B4,5)v0
Theorem 6.4.8. (Pn �K1)v is SDC.
Proof. Let V (Pn�K1) = {uj, vj : 1 ≤ j ≤ n}, where vj are pendant vertices and uj
are vertices of path Pn ;j = 1, 2, . . . , n.
E(Pn �K1) = {ujuj+1 : 1 ≤ j ≤ n− 1}⋃{ujvj : 1 ≤ j ≤ n}.Let (Pn �K1)v denote the graph constructed from switching of arbitrary vertex v
in Pn �K1.
According to different degrees of vertices of (Pn � K1)v, it is required to discuss
following three cases.
Case 1: deg(v1) = 1.
WLOG, let us assume that v1 is the switched pendant vertex.
174
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
In this case it is to be noted that, |V (Pn�K1)v1| = 2n and |E(Pn�K1)v1 | = 4n−4.
Consider a bijection f from V ((Pn �K1)v1) to {1, 2, . . . 2n} defined as below.
f(uj) = 2j − 1; 1 ≤ j ≤ n.
f(vk) = 2k; 1 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n− 2 = ef (0).
Case 2: deg(u1) = 2.
WLOG, let us assume that the switched vertex is u1.
In this case it is to be noted that, |V (Pn�K1)u1)| = 2n and |E(Pn�K1)u1)| = 4n−6.
Consider a bijection f from V (Pn �K1)u1) to {1, 2, . . . 2n} defined as below.
f(u1) = 2.
f(v1) = 1.
f(uj) = 2j − 1; 2 ≤ j ≤ n.
f(vk) = 2k; 2 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n− 3 = ef (0).
Case 3: deg(u2) = 3.
WLOG, let us assume that the switched vertex is u2.
In this case it is to be noted that, |V (Pn�K1)u2)| = 2n and |E(Pn�K1)u2)| = 4n−8.
Consider a bijection f from V (Pn �K1)u2) to {1, 2, . . . 2n} defined as below.
f(u1) = 3.
f(u2) = 2.
f(v1) = 4.
f(v2) = 1.
175
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
f(uj) = 2j − 1; 3 ≤ j ≤ n.
f(vk) = 2k; 3 ≤ k ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n− 4 = ef (1).
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Pn �K1)v is SDC.
Example 6.4.10. Comb P5 �K1 and SDC labeling in (P5 �K1)v1, where (v1) = 1
are demonstrated in the following Figure 6.36.
u1 u2 u3 u4 u5
v1 v2 v3 v4 v5
1 3 5 9
2 4 6 8 10
7
Figure 6.36: SDC labeling in (P5 �K1)v1
Example 6.4.11. Comb P5 �K1 and SDC labeling in (P5 �K1)u1, where (u1) = 2
are demonstrated in the following Figure 6.37.
u1 u2 u3 u4 u5
v1 v2 v3 v4 v5 1
2 3 5 97
4 6 8 10
Figure 6.37: SDC labeling in (P5 �K1)u1
Example 6.4.12. Comb P5 �K1 and SDC labeling in (P5 �K1)u2, where (u2) = 3
are demonstrated in the following Figure 6.38.
176
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
u1 u2 u3 u4 u5
v1 v2 v3 v4 v5
2
1
973 5
4 6 8 10
Figure 6.38: SDC labeling in (P5 �K1)u2
Theorem 6.4.9. (Cn �K1)v is SDC.
Proof. Let V (Cn�K1) = {vj, uj | 1 ≤ j ≤ n}, where vj are pendant vertices and uj
are vertices of degree 3, j = 1, 2, . . . , n.
E(Cn �K1) = {ujuj+1, unu1 | 1 ≤ j ≤ n− 1},⋃{ujvj | 1 ≤ j ≤ n}.Let (Cn�K1)v be the graph constructed from switching of an arbitrary vertex v in
Cn �K1.
Corresponding to the vertices of different degree in Cn�K1, it is required to discuss
following two cases.
Case 1: deg(v1) = 1.
WLOG, let us assume that the switched pendant vertex is v1.
In this case it is to be noted that, |V (Cn�K1)v1| = 2n and |E(Cn�K1)v1 | = 4n−3.
Consider a bijection f from V ((Cn �K1)v1) to {1, 2, . . . 2n} defined as below.
f(vj) = 2j − 1; 1 ≤ j ≤ n.
f(uj) = 2j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n− 1, ef (1) = 2n− 2.
Case 2: deg(u1) = 3.
WLOG, let us assume that the switched vertex is u1.
In this case it is to be noted that, |V (Cn�K1)u1 | = 2n and |E(Cn�K1)u1| = 4n−7.
Consider a bijection f from V ((Cn �K1)u1) to {1, 2, . . . 2n} defined as below.
f(u1) = 1.
f(v1) = 2.
177
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
f(uj) = 2j; 2 ≤ j ≤ n.
f(vj) = 2j − 1; 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 2n− 3, ef (0) = 2n− 4.
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (Cn �K1)v is SDC.
Example 6.4.13. Crown C7�K1 and SDC labeling in (C7�K1)v1, where (v1) = 1
are demonstrated in the following Figure 6.39.
v4
u1
v7
u7
v3
u6
u5u4
u3
v2
u2
v6
v5
v1
v1v7
v6
v5
v4
v3
v2
u1
u7
u6
u5u4
u3
u2
1
2
4
6
8 10
12
14
3
5
7
9
11
13
Figure 6.39: SDC labeling in (C7 �K1)v1
Example 6.4.14. Crown C7�K1 and SDC labeling in (C7�K1)u1, where (u1) = 3
are demonstrated in the following Figure 6.40.
13
v4
u1
v7
u7
v3
u6
u5u4
u3
v2
u2
v6
v5
v1
v7
v6
v5
v4
v3
v2
u7
u6
u5u4
u3
u2
1
2
4
6
8 10
12
145
7
9
11
3v1
u1
Figure 6.40: SDC labeling in (C7 �K1)u1
178
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
Theorem 6.4.10. (ACn)v is SDC, where ACn = Cn � P2.
Proof. Let V (ACn) = {vj, wj, uj | 1 ≤ j ≤ n}, where vj, wj and uj are vertices of
degree 1, 2 and 3 respectively; j = 1, 2, . . . , n.
E(ACn) = {ujuj+1, unu1 | 1 ≤ j ≤ n− 1}⋃{ujwj, wjvj | 1 ≤ j ≤ n}.Let (ACn)v denote the graph constructed from switching of an arbitrary vertex v in
ACn.
According to different degrees of vertices of the graph (ACn)v, it is required to
discuss following three cases.
Case 1: deg(v1) = 1.
WLOG, let us assume that the switched pendant vertex is v1.
In this case it is to be noted that, |V ((ACn)v1)| = 3n and |E((ACn)v1)| = 6n− 3.
Consider a bijection f from V ((ACn)v1) to {1, 2, . . . , 3n} defined as below.
f(v1) = 1.
f(vj) = 2n+ j; 2 ≤ j ≤ n.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
f(wj) = 2j; 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
ef (1) = 3n− 1, ef (1) = 3n− 2.
Case 2: deg(w1) = 2.
WLOG, let us assume that the switched vertex is w1.
In this case it is to be noted that, |V ((ACn)w1)| = 3n and |E((ACn)w1)| = 6n− 5.
Consider a bijection f from V ((ACn)w1) to {1, 2, . . . 3n} defined as below.
f(v1) = 2n.
f(w1) = 1.
f(uj) = 2j + 1; 1 ≤ j ≤ n.
f(wj) = 2(j − 1); 2 ≤ j ≤ n.
f(vj) = 2n+ j; 2 ≤ j ≤ n.
179
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
By looking into the above prescribed pattern,
ef (1) = 3n− 2, ef (0) = 3n− 3.
Case 3: deg(u1) = 3.
WLOG, let us assume that the switched vertex is u1.
In this case it is to be noted that, |V ((ACn)u1)| = 3n and |E((ACn)u1)| = 6n− 7.
Consider a bijection f from V ((ACn)u1) to {1, 2, . . . 3n} defined as below.
For n ≤ 7 :
Subcase 1: n ≡ 0, 2(mod 4)
f(u1) = 1.
f(uj) = 2n+ j; 2 ≤ j ≤ n.
f(w2j−1) = 4j − 2; 1 ≤ j ≤ n
2.
f(w2j) = 4j − 1; 1 ≤ j ≤ n
2.
f(v2j−1) = 4j; 1 ≤ j ≤ n
2.
f(v2j) = 4j + 1; 1 ≤ j ≤ n
2.
Subcase 2: n ≡ 1, 3(mod 4)
f(u1) = 1.
f(u2) = 2n+ 1.
f(u3) = 2n+ 3.
f(uj) = 2n+ j; 4 ≤ j ≤ n.
f(w2j−1) = 4j − 2; 1 ≤ j ≤⌈n
2
⌉.
f(w2j) = 4j − 1; 1 ≤ j ≤⌊n
2
⌋.
f(v2j−1) = 4j; 1 ≤ j ≤⌈n
2
⌉.
f(v2j) = 4j + 1; 1 ≤ j ≤⌊n
2
⌋.
180
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
By looking into the above prescribed pattern,
ef (1) =
⌊6n− 5
2
⌋, ef (0) =
⌈6n− 5
2
⌉.
For n > 7 :
Subcase 1: n ≡ 0, 2(mod 4)
f(u1) = 1.
f(u2) = 2n+ 3.
f(u3) = 2n+ 2.
f(uj) = 2n+ j; 4 ≤ j ≤ n.
f(w2j−1) = 4j − 2; 1 ≤ j ≤ n
2.
f(w2j) = 4j − 1; 1 ≤ j ≤ n
2.
f(v2j−1) = 4j; 1 ≤ j ≤ n
2.
f(v2j) = 4j + 1; 1 ≤ j ≤ n
2.
Subcase 2: n ≡ 1, 3(mod 4)
f(u1) = 1.
f(u2) = 2n+ 1.
f(u3) = 2n+ 4.
f(u4) = 2n+ 3.
f(uj) = 2n+ j; 4 ≤ j ≤ n.
f(w2j−1) = 4j − 2; 1 ≤ j ≤⌈n
2
⌉.
f(w2j) = 4j − 1; 1 ≤ j ≤⌊n
2
⌋.
f(v2j−1) = 4j; 1 ≤ j ≤⌈n
2
⌉.
f(v2j) = 4j + 1; 1 ≤ j ≤⌊n
2
⌋.
181
6.4. SDC Labeling With the Use of Switching of a Vertex in Wheel and Shell Allied Graphs
By looking into the above prescribed pattern,
ef (0) =
⌊6n− 5
2
⌋, ef (1) =
⌈6n− 5
2
⌉.
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, (ACn)v is SDC.
Example 6.4.15. Armed crown AC5 and SDC labeling in (AC5)v1, where deg(v1) =
1 are demonstrated in the following Figure 6.41.
u2
v1
v5
u1
u5
u4u3
v4
v3
v2
w1
w4
w2
w3
1
7 9
3
11
2
10
86
4 5
13
14
15
12
w5u2
u1
u5
u4
u3
v1
v5
v4
v3
v2
w1
w4
w2
w3
w5
Figure 6.41: SDC labeling in (AC5)v1
Example 6.4.16. Armed crown AC5 and SDC labeling in (AC5)w1, where deg(w1) =
2 are demonstrated in the following Figure 6.42.
1
7 9
3
11 58
64
2
10
13
14
15
12
w1
u2
u1
u5
u4
u3
w4
w2
w3
w5
v1
v5
v4
v3
v2
u2
v1
v5
u1
u5
u4u3
v4
v3
v2
w1
w4
w2
w3
w5
Figure 6.42: SDC labeling in (AC5)w1
Example 6.4.17. Armed crown AC5 and SDC labeling in (AC5)u1, where deg(u1) =
3 are demonstrated in the following Figure 6.43.
182
6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph
u2
v1
v5
u1
u5
u4u3
v4
v3
v2
w1
w4
w2
w3
w5
1
8
6
10
12
15
24
79
5
13 14
3
11
w1
u2
u1
u5
u4
u3
w4
w2
w3
w5
v1
v5
v4
v3
v2
Figure 6.43: SDC labeling in (AC5)u1
6.5 SDC Labeling by Duplicating a Vertex/Edge in Star
Graph
Vaidya and Prajapati[55] derived some attractive results on prime labeling of graphs
constructed by duplicating the graph elements. In this segment we demonstrate
some SDC graphs constructed by duplicating vertex/edge in star K1,n.
Theorem 6.5.1. The graph constructed by duplicating any vertex in star K1,n is
SDC graph.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)
are pendant vertices of K1,n and E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating any vertex vj by vertex v′j in
K1,n.
Corresponding to the vertices of different degree in K1,n, it is required to discuss
following two cases.
Case 1: deg(v0) = n (duplicating apex vertex).
The graph constructed by duplicating apex vertex v0 in K1,n is the graph K2,n and
hence it is SDC graph (Refer [44]).
Case 2: deg(vj) = 1 (duplicating pendant vertex).
The graph constructed by duplicating any pendant vertex in K1,n is a star graph
K1,n+1 and hence it is SDC graph (Refer [44]).
Theorem 6.5.2. The graph constructed by duplicating any edge in star K1,n is SDC.
183
6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)
are pendant vertices of K1,n.
E(K1,n) = {v0vi | 1 ≤ i ≤ n}.Let G denote the graph constructed by duplicating an edge, say v0vn by a new edge
v′0v′n in K1,n.
Hence in G, d(v0) = n, d(v′0) = n, d(vn) = 1, d(v′n) = 1 and d(vj) = 2, 1 ≤ j ≤ n− 1.
It is to be noted that, |V (G)| = n+ 3 and |E(G)| = 2n.
Consider a bijection f from V (G) to {1, 2, . . . n+ 3} defined as below.
f(v0) = 1.
f(vn) = 3.
f(v′0) = 2.
f(v′n) = 5.
f(v1) = 4.
f(vj) = 4 + j; 2 ≤ j ≤ n− 1.
By looking into the above prescribed pattern, ef (1) = n = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating any edge in K1,n is SDC graph.
Example 6.5.1. The star graph K1,8 and SDC labeling in the graph constructed by
duplicating edge v0v8 in K1,8 are demonstrated in the following Figure 6.44.
12
4
6
7
8 3
11
5v0
v1
v2
v3
v4 v5
v6
v8
v7
v0 v8
v7
v1
v2
v3
v4
v5
v6
v'0v'8
10
9
Figure 6.44: SDC labeling in the graph constructed by duplicating edge in K1,8
184
6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph
Theorem 6.5.3. The graph constructed by duplicating a vertex by an edge in star
K1,n is SDC graph.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)
are pendant vertices of K1,n.
E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating a vertex vj by edge v′jv
′′j in K1,n.
Corresponding to the vertices of different degree in K1,n, it is required to discuss
following two cases.
Case 1: deg(v0) = n.
Let us duplicate apex vertex v0 by an edge v′0v′′0 .
It is to be noted that, |V (G)| = n+ 3 and |E(G)| = n+ 3.
Consider a bijection f from V (G) to {1, 2, 3, . . . , n+ 3} defined as below.
f(v0) = 1.
f(v1) = 3.
f(v′0) = 4.
f(v′′0) = 2.
f(vj) = 3 + j; 2 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
n ≡ 1, 3(mod 4) ef (1) = n+32
= ef (0)
Case 2: deg(vj) = 1.
Let us duplicate pendant vertex vj by an edge v′jv′′j .
WLOG, assume that vj = vn. Then the vertices vn, v′n and v′′n produce a cycle in G.
It is to be noted that, |V (G)| = n+ 3 and |E(G)| = n+ 3.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2, n+ 3} defined as below.
f(v0) = 1.
f(vn) = 3.
185
6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph
f(v′n) = 4.
f(v′′n) = 2.
f(vj) = 4 + j; 1 ≤ j ≤ n− 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
n ≡ 1, 3(mod 4) ef (1) = n+32
= ef (0)
Then we get, |ef (0)− ef (1)| ≤ 1 in each case.
That is, the graph constructed by duplicating vertex by edge in K1,n is SDC.
Example 6.5.2. The star graph K1,5 and SDC labeling in the graph constructed by
duplicating apex vertex v0 by edge v′0v′′0 in K1,5 are demonstrated in the following
Figure 6.45.
v0
v0'
v0"
v1
v2
v3
v4
v5
1
3
2
5
4
7
6
8
v0
v1
v2
v3 v4
v5
Figure 6.45: SDC labeling in the graph constructed by duplicating apex vertex v0 by edge v′0v′′0 in
K1,5
Example 6.5.3. The star graph K1,7 and SDC labeling in the graph constructed by
duplicating vertex v7 by edge v′7v′′7 in K1,7 are demonstrated in the following Figure
6.46.
v4 v5
v6
v7
v0
v1
v2
v3
v4 v5
v6
v7 v7'
v7"
1
3
2
5
4
7
6
8 9
10
v0
v1
v2
v3
Figure 6.46: SDC labeling in the graph constructed by duplicating vertex v7 by edge v′7v′′7 in K1,7
186
6.5. SDC Labeling by Duplicating a Vertex/Edge in Star Graph
Theorem 6.5.4. The graph constructed by duplicating any arbitrary edge by a vertex
in star K1,n is SDC graph.
Proof. Let V (K1,n) = {vj | 0 ≤ j ≤ n}, where v0 is apex vertex and vj(1 ≤ j ≤ n)
are pendant vertices.
E(K1,n) = {v0vj | 1 ≤ j ≤ n}.Let G denote the graph constructed by duplicating an edge say v0vn by a vertex v′n.
It is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 2.
Consider a bijection f from V (G) to {1, 2, . . . , n+ 2} defined as below.
f(v0) = 1.
f(v1) = 2.
f(v′1) = 4.
f(vj) = 2j − 1. 2 ≤ j ≤ 3
f(vj) = 2 + j; 4 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) = n+22
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌊n+22
⌋, ef (1) =
⌈n+22
⌉
Thus |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating edge by vertex in K1,n is SDC.
Example 6.5.4. The star graph K1,6 and SDC labeling in the graph constructed
by duplicating the edge v0v1 by vertex v′1 in K1,6 are demonstrated in the following
Figure 6.47.
187
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
v0
v'1
v1
v2
v3
v4
v5
1
3
2
5
4
7
6
8
v6
v0
v1v2
v3
v4 v5
v6
Figure 6.47: SDC labeling in the graph constructed by duplicating an edge by a vertex in K1,6
6.6 SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
Vaidya and Barasara[54] derived some attractive results on geometric mean label-
ing of graphs constructed by duplicating the graph elements. In this segment we
demonstrate some divisor cordial graphs constructed by duplicating vertex/edge in
cycle Cn.
Theorem 6.6.1. The graph constructed by duplicating an arbitrary vertex in cycle
Cn is SDC graph.
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let us assume that vertex v1 is duplicated by vertex v′1.
Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n+ 2.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′1) =
n+ 1 ; if n ≡ 0, 1, 3(mod 4).
n ; if n ≡ 2(mod 4).
By looking into the above prescribed pattern,
188
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (1) = n+22
= ef (0)
n ≡ 1, 3(mod 4) ef (1) =⌊n+22
⌋, ef (0) =
⌈n+22
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating an arbitrary vertex in Cn is SDC.
Example 6.6.1. The cycle graph C5 and SDC labeling in the graph constructed by
duplicating vertex v1 by vertex v′1 in C5 are demonstrated in the following Figure
6.48.
v5
v4v3
v2
v11
3
2 4
5
6v1'
v2
v3v4
v5
v1
Figure 6.48: SDC labeling in the graph constructed by duplicating a vertex in C5
Theorem 6.6.2. The graph constructed by duplicating an arbitrary edge in cycle
Cn is SDC graph.
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let us assume that v1v2 be the duplicated edge.
By the effect of this duplication, let v′1 and v′2 be lately inserted vertices such that
N(v′1) = {v′2, vn} and N(v′2) = {v′1, v3}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 3.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2} defined as below.
Case 1: n ≡ 0(mod 4).
f(v′1) = n+ 1.
f(v′2) = n+ 2.
189
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
Case 2: n ≡ 2(mod 4).
f(v′1) = n+ 2.
f(v′2) = n.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n+ 1.
Case 3: n ≡ 1, 3(mod 4).
f(v′1) = 1.
f(v′2) = 3.
f(vj) =
j + 1 ; j ≡ 1(mod 4)
j + 2 ; j ≡ 2, 3(mod 4)
j + 3 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 2.
f(vn−1) =
n+ 1 ; if n ≡ 1(mod 4).
n+ 2 ; if n ≡ 3(mod 4).
f(vn) =
n+ 2 ; if n ≡ 1(mod 4).
n+ 1 ; if n ≡ 3(mod 4).
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
n ≡ 1, 3(mod 4) ef (1) = n+32
= ef (0)
190
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating an arbitrary edge in Cn is SDC.
Example 6.6.2. The cycle graph C6 and SDC labeling in the graph constructed by
duplicating edge v1v2 in C6 are demonstrated in the following Figure 6.49.
v6
v5
v4
v1
v2
v3
3
12
4
5
7
6
8
v2'
v1'
v6
v5
v4
v3
v2
v1
Figure 6.49: SDC labeling in the graph constructed by duplicating an edge in C6
Theorem 6.6.3. The graph constructed by duplicating any arbitrary vertex by a
new edge in cycle Cn is SDC graph for n ≡ 0, 1, 2(mod 4).
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n} and E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let v1 be the vertex duplicated by edge v′1v
′2.
Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 3.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′1) =
n+ 1 ; if n ≡ 0, 1(mod 4).
n ; if n ≡ 2(mod 4).
f(v′2) = n+ 2.
By looking into the above prescribed pattern,
191
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
Cases of n Edge label conditions
n ≡ 0(mod 4) ef (1) =⌊n+32
⌋, ef (0) =
⌈n+32
⌉
n ≡ 1(mod 4) ef (1) = n+32
= ef (0)
n ≡ 2(mod 4) ef (0) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating any arbitrary vertex by a new edge
in K1,n is SDC.
Example 6.6.3. The cycle graph C5 and SDC labeling in the graph constructed by
duplicating vertex v1 by new edge v′1v′2 in C5 are demonstrated in the following Figure
6.50.
v1
v5
v4v3
v2
v1' v2'
1
3
2 4
5
6 7
v1
v5
v4v3
v2
Figure 6.50: SDC labeling in the graph constructed by duplicating vertex by edge in C5
Theorem 6.6.4. The graph constructed by duplicating each vertex by new edge in
cycle Cn is SDC graph.
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n}.E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.Let G denote the graph constructed by duplicating each vertex by new edge in cycle
Cn.
Also it is to be noted that, |V (G)| = 3n and |E(G)| = 4n.
Consider a bijection f from V (G) to {1, 2, 3, . . . 3n} defined as below.
192
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
For n ≡ 0, 1, 3(mod 4) :
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
Case 1: n ≡ 0, 3(mod 4).
f(v′j) =
2j − 1 + n ; j ≡ 1, 3(mod 4)
2j − 2 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌈n2
⌉.
f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌈n
2
⌉.
f(v′j) = n+ 2j − 1;⌈n
2
⌉+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;⌈n
2
⌉+ 1 ≤ j ≤ n.
Case 2: n ≡ 1(mod 4).
f(v′j) =
2j + n ; j ≡ 1, 3(mod 4)
2j − 3 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌊n2
⌋.
f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌊n
2
⌋.
f(v′j) = n+ 2j − 1;⌊n
2
⌋+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;⌊n
2
⌋+ 1 ≤ j ≤ n.
Case 3: n ≡ 2(mod 4).
f(vj) =
j + 1 ; j ≡ 1(mod 4)
j + 2 ; j ≡ 2(mod 4)
j − 2 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n.
f(v′1) = n− 1.
f(v′′1) = n+ 1.
193
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
f(v′j) =
2j − 3 + n ; j ≡ 1, 3(mod 4)
2j + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n2.
f(v′′j ) = f(v′j) + 2; 2 ≤ j ≤ n
2.
f(v′j) = n+ 2j − 1;n
2+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;n
2+ 1 ≤ j ≤ n.
Due to the above prescribed pattern, we have
ef (1) = 2n = ef (0).
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating each vertex by new edge in cycle Cn
is SDC.
Example 6.6.4. The cycle graph C5 and SDC labeling in the graph constructed by
duplicating each vertex by new edge in cycle C5 are demonstrated in the following
Figure 6.51.
v3
v5
v1
v2
v4
v1'
v5'
v1"
v5"
v4'v3"
v3'
v2"
v2'
v4"
1
3
2 4
56
7
8
11
9
12
13
14
15
10
v3
v5
v1
v2
v4
Figure 6.51: SDC labeling in the graph constructed by duplicating each vertex by new edge incycle C5
Theorem 6.6.5. The graph constructed by duplicating any arbitrary edge by a new
vertex in cycle Cn is SDC graph.
Proof. Let V (Cn) = {vj | 1 ≤ j ≤ n} and E(Cn) = {vjvj+1, vnv1 | 1 ≤ j ≤ n− 1}.WLOG, let v1vn be the edge which is duplicated by new vertex v′.
194
6.6. SDC Labeling by Duplicating Vertex/Edge in Cycle Graph
Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n+ 2.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.
Case 1: n ≡ 0, 1, 3(mod 4).
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′) = n+ 1.
Case 2: n ≡ 2(mod 4).
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n− 1.
f(vn) = n+ 1.
f(v′) = n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 1(mod 4) ef (1) = n+32
= ef (0)
n ≡ 2(mod 4) ef (0) =⌊n+32
⌋, ef (1) =
⌈n+32
⌉
n ≡ 3(mod 4) ef (1) =⌊n+32
⌋, ef (0) =
⌈n+32
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating any arbitrary edge by new vertex in
Cn is SDC.
Example 6.6.5. The cycle graph C7 and SDC labeling in the graph constructed by
duplicating edge v1v2 by new vertex v′ in C7 are demonstrated in the following Figure
6.52.
195
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
v7
v6v5
v4
v3
v2
v1
v'
1
3
2
4 5
6
7
8
v6
v5v4
v3
v2
v1
v7
Figure 6.52: SDC labeling in the graph constructed by duplicating edge by new vertex in C7
6.7 SDC Labeling by duplicating Vertex/Edge in Path Graph
In the previous segment, SDC graphs using duplicating vertex/edge in cycle Cn
are derived while in this segment we demonstrate some divisor cordial graphs con-
structed by duplicating vertex/edge in path Pn.
Theorem 6.7.1. The graph constructed by duplicating an arbitrary vertex in path
Pn is SDC graph.
Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and
vj(2 ≤ j ≤ n− 1) are internal vertices.
To label the vertices in the graph constructed by duplicating vertex in Pn, we need
to consider the following two cases.
Corresponding to the vertices of different degree in Pn, it is required to discuss
following two cases.
Case 1: deg(v1) = 1.
WLOG let the pendant vertex v1 be duplicated by new vertex v′1.
Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n.
Consider a bijection f from V (G) to {1, 2, 3, . . . , n+ 1} as follows.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′1) = n+ 1.
196
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
Case 2: deg(vk) = 2.
WLOG let v3 be the vertex of degree 2 which is duplicated by new vertex v′3.
It is to be noted that, |V (G)| = n+ 1 and |E(G)| = n+ 1.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′3) = n+ 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0(mod 4) ef (0) =⌊n2
⌋, ef (1) =
⌈n2
⌉
n ≡ 1, 2, 3(mod 4) ef (1) =⌊n2
⌋, ef (0) =
⌈n2
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating any arbitrary vertex in Pn is SDC.
Example 6.7.1. SDC labeling in the graph constructed by duplicating pendant vertex
v1 and vertex v3 (of degree 2) in P5 are demonstrated in the following Figure 6.53.
v1 v2 v3 v4 v5
1 3 2 4 5
6 v3'
v1 v2 v3 v4 v5
1 3 2 4 5
6 v1'
Figure 6.53: SDC labeling in the graph constructed by duplicating a vertex in P5
Theorem 6.7.2. The graph constructed by duplicating an arbitrary edge in path Pn
is SDC graph.
Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and
vj(2 ≤ j ≤ n− 1) are internal vertices.
197
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
Let E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.In Pn, there are two types of edges.
(1) Pendant edges (whose one end vertex is pendant vertex).
(2) Internal edges.
To label the vertices in the graph constructed by duplicating edge in Pn, we need to
consider following two cases.
Case 1: Duplicating a pendant edge.
WLOG, let v1v2 be the duplicated edge.
Let v′1 and v′2 be lately inserted vertices due to the duplicating such that N(v′1) =
{v′2} and N(v′2) = {v′1, v3}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 1.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′1) = n+ 1.
f(v′2) = n.
Case 2: Duplicating an internal edge.
Let v2v3 be duplicated internal edge.
Let v′2 and v′3 be lately inserted vertices due to the duplicating such that N(v′2) =
{v′3, v1} and N(v′3) = {v′2, v4}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 2.
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′2) = n+ 1.
f(v′3) = n.
198
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0(mod 4) ef (0) =⌊n+12
⌋, ef (1) =
⌈n+12
⌉
n ≡ 1, 2, 3(mod 4) ef (1) =⌊n+12
⌋, ef (0) =
⌈n+12
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating an arbitrary edge in Pn is SDC.
Example 6.7.2. SDC labeling in the graph constructed by duplicating pendant edge
v1v2 and duplicating internal edge v2v3 in P5 are demonstrated in the following Figure
6.54.
v1 v2 v3 v4 v5
1 3 2 4 5
7 6
v1 v2 v3 v4 v5
1 3 2 4 5
7 6
Figure 6.54: SDC labeling in the graph constructed by duplicating edge in P5
Theorem 6.7.3. The graph constructed by duplicating an arbitrary vertex by a new
edge in path Pn is SDC graph.
Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and
vj(2 ≤ j ≤ n− 1) are internal vertices.
E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.WLOG, let vk be the vertex duplicated by edge v′kv
′′k .
Let v′k and v′′k be lately inserted vertices due to the duplicating such that N(v′k) =
{v′′k , vk} and N(v′′k) = {v′k, vk}.Also it is to be noted that, |V (G)| = n+ 2 and |E(G)| = n+ 2.
199
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 2} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′k) = n.
f(v′′k) = n+ 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) = n+22
= ef (1)
n ≡ 1, 3(mod 4) ef (1) =⌊n+22
⌋, ef (0) =
⌈n+22
⌉
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating an arbitrary vertex in Pn is SDC.
Example 6.7.3. SDC labeling in the graph constructed by duplicating vertex v′2 by
a new edge v′2v′′2 in P5 is demonstrated in the following Figure 6.55.
v1 v2 v3 v4 v5
1 3 2 4 5
v2' v2"
76
Figure 6.55: SDC labeling in the graph constructed by duplicating vertex v′2 by a new edge v′2v′′2
in P5
Theorem 6.7.4. The graph constructed by duplicating an arbitrary edge by a new
vertex in path Pn is SDC graph.
Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and
vj(2 ≤ j ≤ n− 1) are internal vertices.
E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.WLOG, let vkvk+1 be the duplicated edge and v′ be the lately inserted vertex due
to this duplicating.
Also it is to be noted that, |V (G)| = n+ 1 and |E(G)| = n+ 1.
200
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
Consider a bijection f from V (G) to {1, 2, 3, . . . n+ 1} defined as below.
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
f(v′) = n+ 1.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0, 2(mod 4) ef (0) =⌊n+12
⌋, ef (1) =
⌈n+12
⌉
n ≡ 1, 3(mod 4) ef (0) = n+12
= ef (1)
Then we get, |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating an arbitrary edge by a new vertex in
Pn is SDC.
Example 6.7.4. SDC labeling in the graph constructed by duplicating edge v2v3 by
a new vertex v′ in P5 is demonstrated in the following Figure 6.56.
v1 v2 v3 v4 v5
1 3 2 4 5
6
Figure 6.56: SDC labeling in the graph constructed by duplicating an edge v2v3 by a new vertexv′ in P5
Theorem 6.7.5. The graph constructed by duplicating each vertex by edge in path
Pn is SDC graph.
Proof. Let V (Pn) = {vj | 1 ≤ j ≤ n}, where v1, vn are pendant vertices and
vj(2 ≤ j ≤ n− 1) are internal vertices.
E(Pn) = {vjvj+1 | 1 ≤ j ≤ n− 1}.Let G denote the graph constructed by duplication each vertex by edge in path Pn.
Also it is to be noted that, |V (G)| = 3n and |E(G)| = 4n− 1.
Let the edge inserted due to duplicating vertex vk has end vertices v′k and v′′k .
201
6.7. SDC Labeling by duplicating Vertex/Edge in Path Graph
Consider a bijection f from V (G) to {1, 2, 3, . . . 3n} defined as below.
For n ≡ 0, 1, 3(mod 4) :
f(vj) =
j ; j ≡ 0, 1(mod 4)
j + 1 ; j ≡ 2(mod 4)
j − 1 ; j ≡ 3(mod 4); 1 ≤ j ≤ n.
For the remaining vertices v′1, v′2, . . . , v
′n, v
′′1 , v′′2 , . . . , v
′′n, let us consider following cases.
Case 1: n ≡ 0, 3(mod 4).
f(v′j) =
2j − 1 + n ; j ≡ 1, 3(mod 4)
2j − 2 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌈n2
⌉.
f(v′′j ) = f(v′j) + 2, 1 ≤ j ≤⌈n
2
⌉.
f(v′j) = n+ 2j − 1;⌈n
2
⌉+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;⌈n
2
⌉+ 1 ≤ j ≤ n.
Case 2: n ≡ 1(mod 4).
f(v′j) =
2j + n ; j ≡ 1, 3(mod 4)
2j − 3 + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤⌊n2
⌋.
f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤⌊n
2
⌋.
f(v′j) = n+ 2j − 1;⌊n
2
⌋+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;⌊n
2
⌋+ 1 ≤ j ≤ n.
Case 3: n ≡ 2(mod 4).
f(vj) =
j + 1 ; j ≡ 1(mod 4)
j + 2 ; j ≡ 2(mod 4)
j − 2 ; j ≡ 3(mod 4)
j − 1 ; j ≡ 0(mod 4); 1 ≤ j ≤ n− 1.
202
6.8. Conclusion and Scope for Further Research
f(v′j) =
2j − 3 + n ; j ≡ 1, 3(mod 4)
2j + n ; j ≡ 0, 2(mod 4); 1 ≤ j ≤ n2.
f(v′′j ) = f(v′j) + 2; 1 ≤ j ≤ n
2.
f(v′j) = n+ 2j − 1;n
2+ 1 ≤ j ≤ n.
f(v′′j ) = n+ 2j;n
2+ 1 ≤ j ≤ n.
By looking into the above prescribed pattern,
Cases of n Edge label conditions
n ≡ 0(mod 4) ef (0) =⌊4n−1
2
⌋, ef (1) =
⌈4n−1
2
⌉
n ≡ 1, 2, 3(mod 4) ef (0) =⌈4n−1
2
⌉, ef (1) =
⌊4n−1
2
⌋
Then we get, in each case |ef (0)− ef (1)| ≤ 1.
That is, the graph constructed by duplicating of each vertex by edge in Pn is SDC.
Example 6.7.5. SDC labeling in the graph constructed by duplicating each vertex
by edge in P5 is demonstrated in the following Figure 6.57.
v1 v2 v3 v4 v5
1 3 2 4 5
7
6
v1' v1"
v2"v2' v4"v4'
v3"v3' v5"v5'
8
119
12 13
14 1510
Figure 6.57: SDC labeling in the graph constructed by duplicating each vertex by edge in P5
6.8 Conclusion and Scope for Further Research
In this chapter we have emanated new SDC graphs which are constructed from the
standard graph families by considering graph operations such as ringsum of different
graphs with star graph, switching of vertex in cycle allied graph and duplicating
vertex in star, path and cycle graphs. At the end, we pose some open problems.
203
6.8. Conclusion and Scope for Further Research
Problem 6.8.1. Classify/Generalize the graphs G such that G⊕K1,n is SDC graph.
(Here |V (G)| may or may not be equal to n.)
Problem 6.8.2. Investigate some new SDC graphs with respect to other graph op-
erations.
Problem 6.8.3. The graph constructed from switching of any vertex (except apex)
of helm Hn is SDC graph.
Problem 6.8.4. The graph constructed from switching of any vertex (except apex)
of closed helm CHn is SDC graph.
204
CHAPTER 7
Summary
This thesis contributes extensive results, novel graphs and connecting concepts of
graph labeling techniques with other field of mathematics such as combinatorics and
number theory.
Labeling of a graph is a bridge connecting combinatorics and graph theory. Large
number of Number Theory results have been used to prove different graph families
which satisfy different graph labeling patterns. During the entire research work du-
ration, 16 new graphs in the field of divisor cordial labeling, 8 new graphs in the
field of square divisor cordial and cube divisor cordial labeling, 24 new graphs in the
field of vertwx odd divisor cordial labeling and 68 new graphs in the field of sum
divisor cordial labeling are discovered. We have derived some algebraic properties of
it. We have also stated open problems and future scope of research in each labeling
technique at the end of each chapter.
205
Details of papers presented in conferences and pub-lished in journals arising from the thesis
Below is the list of research papers published in journals and presented in con-
ferences to obtain the depth of content and to acquire knowledge about the current
ongoing research.
Research Papers Presented in Conferences:
z Research paper entitled “Divisor Cordial Labeling for Vertex Switching and
Duplication of Special Graphs” was presented in National Conference on Al-
gebra, Analysis & Graph Theory [NCAAG - 2017] during 9-11 February 2017,
organized by Department of Mathematics, Saurastra University, Rajkot.
z Research paper entitled “Sum Divisor Cordial Labeling for Vertex Switching
of Cycle Related graphs” was presented in National Conference on Recent Ad-
vancements in Graph Theory, [RAGT - 2019] during 9-10 November 2019, or-
ganized by Department of Mathematics, Gujarat University, Ahmedabad.
z Research paper entitled “Sum Divisor Cordial Labeling in the Context of Corona
Product of graphs” was presented in National Conference on Mathematical Sci-
ences [NCMS - 2020] during 10-11 January 2020, organized by Department of
Mathematics, College of Arts and Science, Adipur, Kachchh.
Research Papers Published in Journals:
1. G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling for Vertex Switch-
ing and Duplication of Special Graphs, International Journal of Mathematics
and its Applications, Volume 4(3A) (2016), 73-80.
2. G. V. Ghodasara and D. G. Adalja, Square Divisor Cordial, Cube Divisor Cor-
dial and Vertex Odd Divisor Cordial Labeling of Graphs, International Journal
of Mathematics Trends and Technology, Volume 39(2) (2016), 118-122.
3. G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling in Context of
Ringsum of Graphs, International Journal of Mathematics and Soft Computing,
Volume 7(3) (2017), 23-31.
206
4. G. V. Ghodasara and D. G. Adalja, Vertex Odd Divisor Cordial Labeling for
Vertex Switching of Special Graphs, Global Journal of Pure and Applied Math-
ematics, Volume 9(13) (2017), 5525-5538.
5. D. G. Adalja and G. V. Ghodasara, Vertex Odd Divisor Cordial Labeling of
Ringsum of Different Graphs with Star Graph, Research & Reviews: Discrete
Mathematical Structures, Volume 5(2) (2018), 1-9.
6. D. G. Adalja and G. V. Ghodasara, Some New Sum Divisor Cordial Graphs,
International Journal of Applied Graph Theory, Volume 2(1) (2018), 19-33.
7. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling of Snakes
Related Graphs, Journal of Computer and Mathematical Sciences, Volume 9(7)
(2018), 754-768.
8. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling of Ring Sum
of a Graph With Star Graph, International Journal of Computer Sciences and
Engineering, Volume 6(5) (2018), 1-7.
9. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling in the Context
of Corona Product of Graphs, Journal of Applied Science and Computations,
Volume 5(10) (2018), 1141-1158.
Research Papers Communicated for Publication in Journals:
1. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling in the Context
of Vertex Switching of Graphs, Malaya Journal of Mathematics.
2. D. G. Adalja and G. V. Ghodasara, Sum Divisor Cordial Labeling for Duplica-
tion of Special Graphs, Mathematics Todays.
207
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[36] P. Lawrence Rozario Raj and R Lawrence Joseph Manoharan, Some results ondivisor cordial labeling of graphs, International Journal of Innovative Science,Engineering and Technology, Volume 1(10), (2014), 226-231.
[37] P. Lawrence Rozario Raj and R Valli, Some new families of divisor cordialgraphs, International Journal of Mathematics Trends and Technology, Volume7(2), (2014), 94-102.
[38] P. Lawrence Rozario Raj and S. Hema Surya, Some New Families of Sum Di-visor Cordial Graphs, International Journal of Mathematics Trends and Tech-nology, Volume 40(2), (2016), 175-179.
[39] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs,SIAM Journal on Algebraic Discrete Methods, Volume 1(4), (1980), 382-404.
[40] R Shankar and R Uma, Square Divisor Cordial Labeling for Some Graphs, In-ternational Journal of Engineering Sciences & Research Technology, Volume5(10), (2016), 726-730.
[41] R. Frucht and F. Harary, On the corona of two graphs, Aequationes Mathemat-icae (Springer), Volume 4(3), (1970), 322-325.
[42] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs,SIAM Journal on Algebraic Discrete Methods, Volume 1, (1980), 382-404.
[43] R. Ponraj, S. S. Narayanan and R. Kala, Difference cordial labeling of graphs,Global Journal of Mathematical Sciences, Theory and Practical, Volume 5(3),(2013), 185-196.
[44] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor cordialgraphs, International Journal of Mathematical Combinatrics, Volume 4, (2011),15-25.
[45] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Special classesof divisor cordial graphs, International Mathematical Forum, Volume 7(35),(2012), 1737- 1749.
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[47] S Murugesan, D Jayaraman and J Shiama, Square divisor cordial graphs, Inter-national Journal of Computer Applications, Volume 64(22), (2013), 1-4.
[48] Sugumaran and Suresh, Cycle Related Vertex Odd Divisor Cordial Labeling ofGraphs, Journal of Computer and Mathematical Sciences, Volume 9(3), (2018),204-211.
[49] S. K. Vaidya and N. H. Shah, Further results on divisor cordial labeling, Annalsof Pure and Applied Mathematics, Volume 4(2), (2013), 150-159.
[50] S. K. Vaidya and N. H. Shah, Some star and bistar related divisor cordial graphs,Annals of Pure and Applied Mathematics, Volume 3(1), (2013), 67-77.
[51] S. K. Vaidya and N. H. Shah, On square divisor cordial graphs, Journal ofScientific Research, Volume 6(3), (2014), 445-455.
[52] S. K. Vaidya and N. H. Shah, Cordial Labeling of Snakes, International Journalof Mathematics And Its Applications, Volume 2(3), (2014), 17-27.
[53] S. K. Vaidya and C. M. Barasara, Edge product cordial labeling of graphs, Jour-nal of Mathematics and Computer Science, Volume 2(5), (2012), 1436-1450.
[54] S. K. Vaidya and C. M. Barasara, geometric mean labeling in the context ofduplication of graph elements, International Journal of Math. Sci. and Engg.Appls, Volume 6(6), (2012), 311-319.
[55] S. K. Vaidya and U. M. Prajapati, Prime labeling in the context of duplicationof graph elements, International Journal of Mathematics and soft Computing,Volume 3(1), (2013), 13-20.
[56] S. Murugesan , D. Jayaraman and J. Shiama, Square divisor cordial graphs,International Journal of Computer Applications, Volume 64(22), (2013), 1-4.
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211
Annexure
212
International Journal of Mathematics and Soft ComputingVol.7, No.1 (2017), 23 - 31. ISSN Print : 2249 - 3328
ISSN Online : 2319 - 5215
Divisor cordial labeling in context of ring sum of graphs
G. V. Ghodasara1, D. G. Adalja2
1H. & H. B. Kotak Institute of ScienceRajkot, Gujarat, India.
gaurang enjoy@yahoo.co.in
2Marwadi Education FoundationRajkot, Gujarat, India.
divya.adalja@marwadieducation.edu.in
Abstract
A graph G = (V,E) is said to have a divisor cordial labeling if there is a bijectionf : V (G) → {1, 2, . . . |V (G)|} such that if each edge e = uv is assigned the label 1 iff(u)|f(v) or f(v)|f(u) and 0 otherwise, then the number of edges labeled with 0 and thenumber of edges labeled with 1 differ by at most 1. If a graph has a divisor cordial labeling,then it is called divisor cordial graph. In this paper we derive divisor cordial labeling of ringsum of different graphs.
Keywords: Divisor cordial labeling, ring sum of two graphs.AMS Subject Classification(2010): 05C78.
1 Introduction
By a graph, we mean a simple, finite, undirected graph. For terms not defined here, we referto Gross and Yellen [3]. For standard terminology and notations related to number theory werefer to Burton [4]. Varatharajan et al.[7] introduced the concept of divisor cordial labeling of agraph. The divisor cordial labeling of various types of graphs are presented in [6, 8]. The briefsummary of definitions which are necessary for the present investigation are provided below.
Definition 1.1. A mapping f : V (G) → {0, 1} is called binary vertex labeling of G and f(v)is called the label of the vertex v of G under f .
Notation 1.2. For an edge e = uv , the induced edge labeling f∗ : E(G)→ {0, 1} is given byf∗(e) = |f(u)− f(v)|. Thenvf (i) := number of vertices of G having label i under f .ef (i) := number of edges of G having label i under f∗.
Definition 1.3. A binary vertex labeling f of a graph G is called cordial labeling if |vf (0) −vf (1)| ≤ 1 and |ef (0)− ef (1)| ≤ 1. A graph G is cordial if it admits cordial labeling.
23
24 G. V. Ghodasara and D. G. Adalja
The concept of cordial labeling was introduced by Cahit[1]. The concept was generalized andextended to k−equitable labeling[2]. There are other labeling schemes with minor variations incordial theme such as the product cordial labeling, total product cordial labeling, prime cordiallabeling and divisor cordial labeling. The present work is focused on divisor cordial labeling.
Definition 1.4. Let G = (V,E) be a simple, finite, connected and undirected graph. Abijection f : V → {1, 2, . . . |V |} is said to be divisor cordial labeling if the induced functionf∗ : E → {0, 1} defined by
f∗(e = uv) =
1; if f(u)|f(v) or f(v)|f(u),0; otherwise,
satisfies the condition |ef (0) − ef (1)| ≤ 1. A graph that admits a divisor cordial labeling iscalled a divisor cordial graph.
Definition 1.5. A chord of a cycle Cn is an edge joining two non-adjacent vertices.
Definition 1.6. Two chords of a cycle are said to be twin chords if they form a triangle withan edge of the cycle Cn.For positive integers n and p with 5 ≤ p+ 2 ≤ n,Cn,p is the graph consisting of a cycle Cn withtwin chords with which the edges of Cn form cycle Cp,C3 and Cn+1−p without chords.
Definition 1.7. A cycle with triangle is a cycle with three chords which by themselves form atriangle.For positive integers p, q, r and n ≥ 6 with p + q + r + 3 = n, Cn(p, q, r) denotes a cycle withtriangle whose edges form the edges of cycles Cp+2, Cq+2, Cr+2 without chords.
Definition 1.8. Ring sum of two graphs G1 = (V1, E1) and G2 = (V2, E2) denoted by G1⊕G2,is the graph G1 ⊕G2 = (V1 ∪ V2, (E1 ∪ E2)− (E1 ∩ E2)).
Remark 1.9. Throughout this paper we consider the ring sum of a graph G with K1,n byconsidering any one vertex of G and the apex vertex of K1,n as a common vertex.
2 Main ResultsTheorem 2.1. Cn ⊕K1,n is a divisor cordial graph for all n ∈ N.
Proof: Let V (Cn ⊕K1,n) = V1 ∪ V2, where V1 = {u1, u2, . . . , un} be the vertex set of Cn andV2 = {v = u1, v1, v2, . . . , vn} be the vertex set of K1,n. Here v1, v2, . . . , vn are pendant verticesand v is the apex vertex of K1,n. Also |V (Cn ⊕K1,n)| = |E(Cn ⊕K1,n)| = 2n.
We define a labeling f : V (Cn ⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:
f(u1) = f(v) = 2, f(v1) = 1,
Divisor cordial labeling in context of ring sum of graphs 25
f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.
According to this pattern the vertices are labeled such that for any edge e = uiui+1 in Cn,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.
Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n.Hence ef (1) = ef (0) = n.
Thus the graph Cn ⊕K1,n admits a divisor cordial labeling and hence Cn ⊕K1,n is a divisorcordial graph.
Example 2.2. A divisor cordial labeling of C5 ⊕K1,5 is shown in Figure 1.
Figure 1: A divisor cordial labeling of C5 ⊕K1,5.
Theorem 2.3. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 4, n ∈ N, where G isthe cycle Cn with one chord forming a triangle with two edges of Cn.
Proof: Let G be the cycle Cn with one chord. Let V (G ⊕ K1,n) = V1 ∪ V2, where V1 is thevertex set of G and V2 is the vertex set of K1,n. Let u1, u2, . . . , un be the successive vertices ofCn and e = u2un be the chord of Cn.
The vertices u1, u2, un form a triangle with the chord e. Let v1, v2, . . . , vn be the pendantvertices, v be the apex vertex ofK1,n. Take v = u1. Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| =2n+ 1.
We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:f(u1) = f(v) = 2, f(v1) = 1,f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.
According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.
26 G. V. Ghodasara and D. G. Adalja
Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n.
Hence ef (1) = n, ef (0) = n+ 1.
Thus |ef (0) − ef (1)| ≤ 1 and the graph G ⊕ K1,n admits divisor cordial labeling. Therefore,G⊕K1,n is a divisor cordial graph.
Example 2.4. A divisor cordial labeling of ring sum of C6 with one chord and K1,6 is shownin Figure 2.
Figure 2: A divisor cordial labeling of ring sum of C6 with one chord and K1,6.
Theorem 2.5. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 5, n ∈ N, where G isthe cycle with twin chords forming two triangles and another cycle Cn−2 with the edges of Cn.
Proof: Let G be the cycle Cn with twin chords, where chords form two triangles and one cycleCn−2. Let V (G⊕K1,n) = V1 ∪ V2.
V1 = {u1, u2, . . . , un} is the vertex set of Cn, e1 = unu2 and e2 = unu3 are the chords of Cn.
V2 = {v = u1, v1, v2, . . . , vn} is the vertex set of K1,n, where v1, v2, . . . , vn are pendant verticesand v = u1 is the apex vertex.
Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 2.
We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:
f(u1) = f(v) = 2, f(v1) = 3, f(un) = 2n− 1, f(un−1) = 1,f(ui) = 2i+ 1; 2 ≤ i ≤ n− 2,f(vj) = 2j; 2 ≤ j ≤ n.
According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.
Also f(v1) | f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n. and f(un−1) | f(un−2). Henceef (0) = n+ 1 = ef (1).
Divisor cordial labeling in context of ring sum of graphs 27
Thus the graph admits a divisor cordial labeling and henceG⊕K1,n is a divisor cordial graph.
Example 2.6. A divisor cordial labeling of ring sum of cycle C7 with twin chords and K1,7 isshown in Figure 3.
Figure 3: A divisor cordial labeling of ring sum of C7 with twin chords and K1,7.
Theorem 2.7. The graph G⊕K1,n is a divisor cordial graph for all n ≥ 6, n ∈ N, where G isa cycle with triangle Cn(1, 1, n− 5).
Proof: Let G be cycle with triangle Cn(1, 1, n − 5). Let V (G ⊕ K1,n) = V1 ∪ V2, whereV1 = {u1, u2, . . . , un} is the vertex set of G and V2 = {v = u1, v1, v2, . . . , vn} is the vertex setof K1,n. Here v1, v2, . . . , vn are the pendant vertices and v is the apex vertex of K1,n.
Let u1, u3 and un−1 be the vertices of triangle formed by edges e1 = u1u3, e2 = u3un−1 ande3 = u1un−1. Also |V (G⊕K1,n)| = 2n and |E(G⊕K1,n)| = 2n+ 3.
We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:
f(u1 = v) = 2, f(v1) = 3, f(u2) = 1,f(ui) = 2i− 1; 3 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.
According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.
Also f(v1) - f(v) and f(v) | f(vj) for each j, 2 ≤ j ≤ n. f(v) | f(u2), f(u2) | f(u3) .
Then |ef (0)− ef (1)| ≤ 1. Hence G⊕K1,n is a divisor cordial graph.
Example 2.8. A divisor cordial labeling of ring sum of cycle C8 with triangle and K1,8 isshown in Figure 4.
28 G. V. Ghodasara and D. G. Adalja
Figure 4: A divisor cordial labeling of ring sum of C8(1, 1, 3) and K1,8.
Theorem 2.9. The graph Pn ⊕K1,n is a divisor cordial graph for all n ∈ N.
Proof: Let V (Pn ⊕K1,n) = V1 ∪ V2, where V1 = {u1, u2, . . . , un} is the vertex set of Pn andV2 = {v = u1, v1, v2, . . . , vn} is the vertex set of K1,n. Here v1, v2, . . . , vn are the pendantvertices and v is the apex vertex. Also |V (G⊕K1,n)| = 2n, |E(G⊕K1,n)| = 2n− 1.
We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n} as follows:
f(u1) = f(v) = 2, f(v1) = 1,f(ui) = 2i− 1; 2 ≤ i ≤ n,f(vj) = 2j; 2 ≤ j ≤ n.
According to this pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1), 1 ≤ i ≤ n− 1.Also f(v) | f(vj) for each j, 1 ≤ j ≤ n.
Then we have ef (1) = n, ef (0) = n − 1. Hence the graph admits a divisor cordial labeling.Theredore, Pn ⊕K1,n is a divisor cordial graph.
Example 2.10. A divisor cordial labeling of P5 ⊕K1,5 is shown in Figure 5.
Figure 5: A divisor cordial labeling of G⊕K1,5.
Definition 2.11. The double fan graph DFn is defined as DFn = Pn + 2K1.
Theorem 2.12. The graph DFn ⊕K1,n is a divisor cordial graph for every n ∈ N.
Divisor cordial labeling in context of ring sum of graphs 29
Proof: Let V (DFn ⊕ K1,n) = V1 ∪ V2, where V1 = {u,w, u1, u2, . . . , un} be the vertex set ofDFn and V2 = {v = w, v1, v2, . . . , vn} be the vertex set of K1,n. Here v1, v2, . . . , vn are pendantvertices and v be the apex vertex of K1,n. Also |V (G⊕K1,n)| = 2n+ 2, |E(G⊕K1,n)| = 4n−1.We define a labeling f : V (G⊕K1,n)→ {1, 2, 3, . . . , 2n+ 2} as follows:
f(w) = f(v) = 1, f(u) = p, where p is largest prime number.f(v1) = 2, f(u1) = 3,f(ui) = 2i; 1 ≤ i ≤ n.
Assign the remaining labels to the remaining vertices v1, v2, . . . , vn in any order.
According to this pattern the vertices are labeled such that for any edge e = uiui+1, f(ui) -f(ui+1) , 1 ≤ i ≤ n − 1. Also f(v) | f(vj) for each j, 1 ≤ j ≤ n. and f(v) | f(ui) for each i,1 ≤ i ≤ n. and f(u) - f(ui) for each i, 1 ≤ i ≤ n.
Then we have ef (1) = 2n, ef (0) = 2n − 1. Hence the graph admits divisor cordial labeling.Therefore, DFn ⊕K1,n is a divisor cordial graph.
Example 2.13. A divisor cordial labeling of DF5 ⊕K1,5 is shown in Figure 6.
Figure 6: A divisor cordial labeling of DF5 ⊕K1,5.
Definition 2.14. The flower fln is the graph obtained from a helm Hn by joining each pendantvertex to the apex of the helm. It contains three types of vertices: an apex of degree 2n, nvertices of degree 4 and n vertices of degree 2.
Theorem 2.15. The graph fln ⊕K1,n is a divisor cordial graph for every n ∈ N.
Proof: Let V (fln⊕K1,n) = V1∪V2, V1 = {u, u1, u2, . . . , un, w1, w2, . . . , wn} be the vertex set offln, where u is the apex vertex, u1, u2, . . . , un are the internal vertices and w1, w2, . . . , wn are theexternal vertices. Let V2 = {v = w1, v1, v2, . . . , vn} be the vertex set ofK1,n, where v1, v2, . . . , vnare pendant vertices and v is the apex vertex of K1,n. Also note that |V (fln ⊕ K1,n)| =3n+ 1, |E(G⊕K1,n)| = 5n.
We define a labeling f : V (fln ⊕K1,n)→ {1, 2, 3, . . . , 3n+ 1} as follows:
30 G. V. Ghodasara and D. G. Adalja
f(u) = 1,f(ui) = 2i+ 1; 1 ≤ i ≤ n,f(wi) = 2i; 1 ≤ i ≤ n.
Assign the remaining labels to the remaining vertices v1, v2, . . . , vn in any order. According tothis pattern the vertices are labeled such that for any edge e = uiui+1 ∈ G,f(ui) - f(ui+1) ,1 ≤ i ≤ n − 1. Further f(u) | f(ui), f(u) | f(wi) for each i, 1 ≤ i ≤ n and f(v) | f(vi) if i isodd, 1 ≤ i ≤ n.
Then |ef (0)− ef (1)| ≤ 1. Hence the graph admits divisor cordial labeling and fln ⊕K1,n is adivisor cordial graph.
Example 2.16. A divisor cordial labeling of the graph fl4 ⊕K1,4 is shown in Figure 7.
Figure 7: A divisor cordial labeling of fl4 ⊕K1,4.
Remark 2.17. In all the above theorems, for the ring sum operation one can consider anyarbitrary vertex of G and by different permutations of the vertex labels provided in the abovedefined labeling pattern one can easily check that the resultant graph is divisor cordial.
Concluing Remarks: The divisor cordial labeling is an invariant of cordial labeling by minorvariation in the definition using divisor of a number. Here we have derived divisor cordialgraphs in context of the operation ring sum of graphs. It is interesting to see whether divisorcordial graphs are invariant under ring sum or any other graph operation or not.
References[1] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Com-
binatoria, 23 (1987), 201-207.
[2] I. Cahit, On cordial and 3-equitable labellings of graphs, Util. Math., 37(1990), 189-198.
[3] J. Gross and J. Yellen, Graph Theory and its applications, CRC Press, 1999.
[4] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, 1990.
Divisor cordial labeling in context of ring sum of graphs 31
[5] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,19, (2015), # DS6.
[6] S. K. Vaidya and N. H. Shah, Further Results on Divisor Cordial Labeling, Annals of Pureand Applied Mathematics, 4(2) (2013), 150-159.
[7] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor Cordial Graphs, In-ternational J. Math. Combin., 4 (2011), 15-25.
[8] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Special Classes of DivisorCordial Graphs, International Mathematical Forum, 7 (35) (2012), 1737- 1749.
Sum Divisor Cordial Labeling in the Context of Corona
Product of Graphs
D. G. Adalja 1, G. V. Ghodasara 2
1 Marwadi Education Foundation,
Rajkot, Gujarat - INDIA
divya.adalja@marwadieducation.edu.in
2 H. & H. B. Kotak Institute of Science,
Rajkot, Gujarat - INDIA
gaurang.enjoy@gmail.com
Abstract
A graph G = (V,E) is said to have a sum divisor cordial labeling if there exists a bijection
f : V (G) → {1, 2, 3, . . . , |V (G)|} such that each edge e = uv is assigned the label 1 if 2 divides
[f(u) + f(v)] and 0 otherwise, then the number of edges labeled with 0 and the number of edges
labeled with 1 differ by at most 1. If a graph admits a sum divisor cordial labeling, then it is
called sum divisor cordial graph. In this paper we have derived the graphs obtained by taking
corona product of K1 with different graphs like star K1,n, complete bipartite graphs K2,n and
K3,n, wheel, helm, flower, fan, double fan, barycentric subdivision of the star K1,n, cycle with
one chord, twin chords and triangle admit sum divisor cordial labeling.
Key words: Sum divisor cordial labeling, Corona product of two graphs.
AMS Subject classification number: 05C78.
1 Introduction
In this paper, by a graph, we mean a simple, finite, undirected graph. For terms and notations
related to graph theory which are not defined here, we refer to Gross and Yellen[6] and for standard
terminology and notations related to number theory we refer to Burton[2].
Remark 1.1. Throughout this paper |V (G)| and |E(G)| denote the cardinality of vertex set and edge
set of graph G respectively.
Rosa[9] introduced grpah labeling as follows.
If the vertices or edges or both of the graph are assigned valued subject to certain conditions, then it
is known as graph labeling.
Varatharajan et al. introduced the concept of divisor cordial labeling of a graph. The definition of
divisor cordial labeling is given below.
Definition 1.1 ([10]). Let G = (V,E) be a simple, finite, connected and undirected graph. A bijection
f : V (G)→ {1, 2, . . . , |V (G)|} is said to be divisor cordial labeling if the induced function f∗ : E(G)→{0, 1} defined by
f∗(e = uv) =
1 if f(u) | f(v) or f(v) | f(u);
0 otherwise.
1
JASC: Journal of Applied Science and Computations
Volume 5, Issue 10, October/2018
ISSN NO: 1076-5131
Page No:1141
satisfies the condition |ef (0)− ef (1)| ≤ 1.
A graph with a divisor cordial labeling is called a divisor cordial graph.
In [10], Varatharajan et al. proved that the graphs such as path, cycle, wheel, star and some
complete bipartite graphs are divisor cordial graphs. They have also derived some special classes of
divisor graphs such as full binary tree, dragon, corona, G ∗K2,n and G ∗K3,n.
Ghodasara and Adalja[5] derived divisor cordial labeling for graphs obtained by ring sum of some
standard graphs with star graph.
Motivated through the concept of divisor cordial labeling, A. Lourdusamy, F. Patrick and J.
Shiama introduced the concept of sum divisor cordial labeling of graphs which is defined as follows.
Definition 1.2 ([7]). Let G = (V,E) be a simple graph, f : V (G) → {1, 2, 3, . . . , |V (G)|} be a
bijection and the induced function f∗ : E(G)→ {0, 1} be defined as
f∗(e = uv) =
1 if 2 | [f(u) + f(v)];
0 otherwise.
Then f is called sum divisor cordial labeling if |ef (0)− ef (1)| ≤ 1.
A graph which admits sum divisor cordial labeling is called sum divisor cordial graph.
In [8], Lourdusamy et al. proved that shadow graph and splitting graph of K1,n, shadow graph,
subdivision graph, splitting graph and degree splitting graph of Bn,n, subdivision graph of ladder,
corona of ladder and triangular ladder with K1, closed helm are sum divisor cordial graphs.
In [1], Adalja and Ghodasara derived some more sum divisor cordial graphs.
Definition 1.3 ([3]). The corona of a graph G with another graph H, denoted as G�H, is the graph
obtained by taking one copy of G and |V (G)| copies of H and joining the ith vertex of G with an edge
to every vertex in the ith copy of H.
Definition 1.4 ([6]). Let G = (V,E) be a graph. Let e = uv be an edge of G and w be a vertex not
in G. The edge e is said to be subdivided when it is replaced by the edges e′ = uw and e′′ = wv.
Definition 1.5 ([6]). If every edge of graph a G is subdivided, then the resulting graph is called
barycentric subdivision of graph G. In other words barycentric subdivision of the graph is obtained by
inserting a vertex of degree 2 into every edge of original graph. The barycentric subdivision of any
graph G is denoted by S(G).
2 Results
Theorem 2.1. K1,n �K1 is a sum divisor cordial graph.
Proof. Let v1, v2, . . . , vn be pendant vertices and v0 be the apex vertex of K1,n.
Let v′0, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph K1,n �K1.
V (K1,n �K1) = V (K1,n) ∪ {v′0, v′1, v′2, . . . , v′n}.E(K1,n �K1) = E(K1,n) ∪ {viv′i; 0 ≤ i ≤ n}.|V (K1,n �K1)| = 2n + 2.
|E(K1,n �K1)| = 2n + 1.
We define labeling f : V (K1,n �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.
f(v0) = 1;
f(v′0) = 2;
f(vi) = 2i + 1 1 ≤ i ≤ n;
f(v′i) = 2i + 2 1 ≤ i ≤ n.
2
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In view of the above defined labeling pattern, we have
ef (1) =
⌊2n + 1
2
⌋,
ef (0) =
⌈2n + 1
2
⌉.
Thus |ef (1)− ef (0)| ≤ 1.
Thus the graph under consideration admits sum divisor cordial labeling.
That is, K1,n �K1 is a sum divisor cordial graph.
Example 2.1. Sum divisor cordial labeling of the graph K1,6 � K1 is shown in Figure 1 as an
illustration for Theorem 2.1.
v1 v2 v3 v4 v5 v6
v2'v1' v3' v4' v5' v6'
v0'
v01
6
3
4 8 1210 14
5 1197 13
2
Figure 1
Theorem 2.2. K2,n �K1 is sum divisor cordial graph except for n ≡ 0(mod4).
Proof. Let W = U ∪ V be the bipartition of vertex set of K2,n, where
U = {u1, u2} and V = {v1, v2, . . . , vn}.Let u′1, u
′2, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph K2,n �K1.
V (K2,n �K1) = V (K2,n) ∪ {u′1, u′2} ∪ {v′1, v′2, . . . , v′n}.E(K2,n �K1) = E(K2,n) ∪ {u1u
′1, u2u
′2} ∪ {viv′i; 1 ≤ i ≤ n}.
|V (K2,n �K1)| = 2n + 4.
|E(K2,n �K1)| = 3n + 2.
We define labeling f : V (K2,n �K1)→ {1, 2, 3, . . . , 2n + 4} as follows.
f(u1) = 1;
f(u2) = 2;
f(vi) = i + 2 1 ≤ i ≤ n.
For n ≡ 1(mod4)
f(u′1) = 2n + 3;
f(u′2) = 2n + 4.
For k = n−12 :
f(v′2i−1) = (n + 2) + (2i− 1) 1 ≤ i ≤ k;
f(v′2i) = (n + 1) + 2i 1 ≤ i ≤ k;
f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.
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For n ≡ 3(mod4)
f(u′1) = 2n + 4;
f(u′2) = 2n + 3.
For k = n+12 :
f(v′2i−1) = (n + 2) + (2i− 1) 1 ≤ i ≤ k;
f(v′2i) = (n + 1) + 2i 1 ≤ i ≤ k;
f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.
For n ≡ 2(mod4)
f(u′1) = 2n + 3;
f(u′2) = 2n + 4.
For k = n+22 :
f(v′2i−1) = (n + 2) + 2i 1 ≤ i ≤ k;
f(v′2i) = (n + 1) + (2i− 1) 1 ≤ i ≤ k;
f(v′i) = (n + 2) + i k + 1 ≤ i ≤ n.
In view of the above defined labeling pattern we have the following.
Cases of n Edge conditions
n ≡ 1(mod 4) ef (1) =⌈3n+2
2
⌉, ef (0) =
⌊3n+2
2
⌋
n ≡ 2(mod 4) ef (1) = 3n+22 = ef (0)
n ≡ 3(mod 4) ef (1) =⌊3n+2
2
⌋, ef (0) =
⌈3n+2
2
⌉
Thus |ef (1)− ef (0)| ≤ 1.
So, K2,n �K1 is a sum divisor cordial graph.
Example 2.2. Sum divisor cordial labeling of K2,5 �K1 is shown in Figure 2 as an illustration for
Theorem 2.2.
v1 v2 v3 v4 v5
v2'v1' v3' v4' v5'
u1'
u2
u2'
u1
3 5
119
7
12
1 2
8
64
10
13 14
Figure 2
Theorem 2.3. K3,n �K1 is sum divisor cordial graph.
Proof. Let W = U ∪ V be the bipartition of vertex set of K2,n, where
U = {u1, u2, u3} and V = {v1, v2, . . . , vn}.4
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Let u′1, u′2, u′3, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph K3,n �K1.
V (K3,n �K1) = V (K3,n) ∪ {u′j , v′i; 1 ≤ j ≤ 3, 1 ≤ i ≤ n}.E(K3,n �K1) = E(K3,n) ∪ {uju
′j , viv
′i; 1 ≤ j ≤ 3, 1 ≤ i ≤ n}.
|V (K3,n �K1)| = 2n + 6.
|E(K3,n �K1)| = 4n + 3.
We define labeling f : V (K3,n �K1)→ {1, 2, 3, . . . , 2n + 6} as follows.
f(ui) = i 1 ≤ i ≤ 3;
f(u′1) = 5;
f(u′2) = 4;
f(u′3) = 6;
f(vi) = 6 + (2i− 1) 1 ≤ i ≤ n;
f(v′i) = 6 + (2i) 1 ≤ i ≤ n.
In view of the above defined labeling pattern we have
ef (1) =
⌈4n + 3
2
⌉,
ef (0) =
⌊4n + 3
2
⌋.
Thus |ef (1)− ef (0)| ≤ 1.
So, K3,n �K1 is a sum divisor cordial graph.
Example 2.3. Sum divisor cordial labeling of K3,7 �K1 is shown in Figure 3 as an illustration for
Theorem 2.3.
v1 v2 v3 v4 v5
v2'v1' v3' v4' v5'
u1'
u2
u2'
u1
v6 v7
v6' v7'
u3
u3'64
8 12 14
1 3
5
1197 13
2
1715 19
1816 2010
Figure 3
Definition 2.1 ([6]). The wheel graph is join of K1 and Cn, denoted as Wn = Cn + K1. The cycle
Cn forms rim, the vertices corresponding rim are called rim vertices and the vertex corresponding to
K1 is called apex (or hub).
Theorem 2.4. Wn �K1 is sum divisor cordial graph.
Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be rim vertices of Wn.
Let v′0, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph Wn �K1.
V (Wn �K1) = V (Wn) ∪ {v′i; 0 ≤ i ≤ n}.E(Wn �K1) = E(Wn) ∪ {viv′i; 0 ≤ i ≤ n}.|V (Wn �K1)| = 2n + 2.
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|E(Wn �K1)| = 3n + 1.
We define labeling f : V (Wn �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.
For n ≡ 0, 2(mod4)
f(v0) = 1;
f(v′0) = 2n + 2;
f(v2i−1) = 4i− 2 1 ≤ i ≤ n
2;
f(v2i) = 4i− 1 1 ≤ i ≤ n
2;
f(v′2i−1) = 4i 1 ≤ i ≤ n
2;
f(v′2i) = 4i + 1 1 ≤ i ≤ n
2.
For n ≡ 1, 3(mod4)
f(v0) = 2n + 2;
f(v′0) = 1;
f(v2i−1) = 4i− 1 1 ≤ i ≤ n− 1
2;
f(v2i) = 4i− 2 1 ≤ i ≤ n− 1
2;
f(v′2i−1) = 4i + 1 1 ≤ i ≤ n− 1
2;
f(v′2i) = 4i 1 ≤ i ≤ n− 1
2;
f(vn) = 2n;
f(v′n) = 2n + 1.
In view of above defined labeling pattern we have the following.
Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) =⌊3n+1
2
⌋, ef (0) =
⌈3n+1
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 3n+12 = ef (0)
Thus |ef (0)− ef (1)| ≤ 1.
Hence Wn �K1 is a sum divisor cordial graph.
Example 2.4. Sum divisor cordial labeling of W7 �K1 is shown in Figure 4 as an illustration for
Theorem 2.4.
v0
v1
v2
v3
v4
v5
v2'
v1'
v3'
v4'
v5'
v6
v7
v6'
v7'
v0'
1
2
34
5
6
7
8
9
10
11
12
13
14
16
15
Figure 4
6
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Definition 2.2 ([6]). The helm Hn is the graph obtained from a wheel Wn by attaching a pendant
edge to each rim vertex. It contains three types of vertices: an apex of degree n, n vertices of degree
4 and n pendant vertices.
Theorem 2.5. Hn �K1 is sum divisor cordial graph.
Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be vertices of degree 4 and u1, u2, . . . , un be the
pendant vertices of the helm Hn.
Let v′0, v′1, v′2, . . . , v
′n, u′1, u′2, . . . , u
′n be the newly added vertices to obtain the graph Hn �K1.
V (Hn �K1) = V (Hn) ∪ {v′0, v′i, u′i; 1 ≤ i ≤ n}.E(Hn �K1) = E(Hn) ∪ {v0v′0, viv′i, uiu
′i; 1 ≤ i ≤ n}.
Hence |V (Hn �K1)| = 4n + 2 and |E(Hn �K1)| = 5n + 1.
We define labeling f : V (Hn �K1)→ {1, 2, 3, . . . , 4n + 2} as follows.
For n ≡ 0, 1(mod4) :
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 1;
f(v′0) = 4n + 2;
f(v′i) = (n + 1) + 2i 1 ≤ i ≤ n;
f(ui) = n + 2i 1 ≤ i ≤ n;
f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.
For n ≡ 2(mod4) :
f(vi) =
i i ≡ 0, 1(mod 4);
i + 1 i ≡ 2(mod 4);
i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′i) = (n + 1) + 2i 2 ≤ i ≤ n;
f(v′1) = n + 2;
f(u1) = n;
f(ui) = n + 2i 2 ≤ i ≤ n;
f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.
For n ≡ 3(mod4) :
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 4n + 2;
f(v′0) = 1;
f(v′i) = (n + 1) + 2i 1 ≤ i ≤ n;
f(ui) = n + 2i 1 ≤ i ≤ n;
f(u′i) = (3n + 1) + i 1 ≤ i ≤ n.
In view of above defined labeling pattern we have the following.
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Cases of n Edge conditions
n ≡ 0, 2(mod 4) ef (1) =⌊5n+1
2
⌋, ef (0) =
⌈5n+1
2
⌉
n ≡ 1, 3(mod 4) ef (1) = 5n+12 = ef (0)
Thus |ef (0)− ef (1)| ≤ 1.
Hence Hn �K1 is a sum divisor cordial graph.
Example 2.5. Sum divisor cordial labeling of H7 �K1 is shown in Figure 5 as an illustration for
Theorem 2.5.
u1u7
u7'
v0
v1
v1'
v2
v2'
v3
v3'v4
v4'
v5v5'
v6
v6'v7
v7'
u2
u2'
u1'
u3
u4
u4'
u5 u3'
u5'
u6u6'
v0'1
3
2
4
5
7
6
8
910
11
12
14
13
15
19
16
17
22
18
20
21
23
26
24
25
27
28
29
30
Figure 5
Definition 2.3 ([4]). The flower graph fln(n ≥ 3) is obtained from helm Hn by joining each pendant
vertex to the central vertex of Hn.
It contains three types of vertices: an apex of degree 2n, n vertices of degree 4 and n vertices of degree
2.
Theorem 2.6. fln �K1 is sum divisor cordial graph.
Proof. Let v0 be the apex vertex and v1, v2, . . . , vn be the vertices of degree 4 and u1, u2, . . . , un be
the vertices of degree 2 in the flower fln.
Let v′0, v′1, v′2, . . . , v
′n, u′1, u′2, . . . , u
′n be the newly added vertices to obtain the graph fln �K1.
V (fln �K1) = V (fln) ∪ {v′0, v′i, u′i, ; 1 ≤ i ≤ n},E(fln �K1) = E(fln) ∪ {v0v′0, viv′i, uiu
′i; 1 ≤ i ≤ n}.
Hence |V (fln �K1)| = 4n + 2 and |E(fln �K1)| = 6n + 1.
We define labeling f : V (fln �K1)→ {1, 2, 3, . . . , 4n + 2} as follows.
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 1;
f(v′0) = 4n + 2;
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For n ≡ 0(mod4)
f(v′4i−3) = 3n + 4i− 1 1 ≤ i ≤ n
4;
f(v′4i−2) = 3n + 4i + 1 1 ≤ i ≤ n
4;
f(v′4i−1) = 3n + 4i− 2 1 ≤ i ≤ n
4;
f(v′4i) = 3n + 4i 1 ≤ i ≤ n
4;
f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n
2;
f(u2i) = n + 4i− 1 1 ≤ i ≤ n
2;
f(u′2i−1) = n + 4i 1 ≤ i ≤ n
2;
f(u′2i) = n + 4i + 1 1 ≤ i ≤ n
2.
For n ≡ 2(mod4)
f(v′4i−3) = 3n + 4i− 3 1 ≤ i ≤ n
4;
f(v′4i−2) = 3n + 4i− 1 1 ≤ i ≤ n
4;
f(v′4i−1) = 3n + 4i 1 ≤ i ≤ n
4;
f(v′4i) = 3n + 4i + 2 1 ≤ i ≤ n
4;
f(u2i−1) = n + 4i− 3 1 ≤ i ≤ n
2;
f(u2i) = n + 4i 1 ≤ i ≤ n
2;
f(u′2i−1) = n + 4i− 1 1 ≤ i ≤ n
2;
f(u′2i) = n + 4i + 2 1 ≤ i ≤ n
2.
For n ≡ 1(mod4)
f(v′4i−3) = 3n + 4i 1 ≤ i ≤ n
4;
f(v′4i−2) = 3n + 4i + 2 1 ≤ i ≤ n
4;
f(v′4i−1) = 3n + 4i− 3 1 ≤ i ≤ n
4;
f(v′4i) = 3n + 4i− 1 1 ≤ i ≤ n
4;
f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n + 1
2;
f(u′2i−1) = n + 4i 1 ≤ i ≤ n + 1
2;
f(u2i) = n + 4i− 1 1 ≤ i ≤ n− 1
2;
f(u′2i) = n + 4i + 1 1 ≤ i ≤ n− 1
2;
f(v′n) = 4n.
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For n ≡ 3(mod4)
f(v′4i−3) = 3n + 4i 1 ≤ i ≤ n + 1
4;
f(v′4i−2) = 3n + 4i + 2 1 ≤ i ≤ n
4;
f(v′4i−1) = 3n + 4i− 3 1 ≤ i ≤ n
4;
f(v′4i) = 3n + 4i− 1 1 ≤ i ≤ n
4;
f(u2i−1) = n + 4i− 2 1 ≤ i ≤ n + 1
2;
f(u′2i−1) = n + 4i 1 ≤ i ≤ n + 1
2;
f(u2i) = n + 4i− 1 1 ≤ i ≤ n− 1
2;
f(u′2i) = n + 4i + 1 1 ≤ i ≤ n− 1
2;
f(v′n−1) = 4n− 2;
f(v′n) = 4n.
In view of above defined labeling pattern we have the following.
ef (1) =
⌈6n + 1
2
⌉,
ef (0) =
⌊6n + 1
2
⌋.
Thus |ef (0)− ef (1)| ≤ 1.
Hence fln �K1 admits sum divisor cordial labeling and hence it is a sum divisor cordial graph.
Example 2.6. Sum divisor cordial labeling of fl7 �K1 is shown in Figure 6 as an illustration for
Theorem 2.6.
v0
v1
v1'
v2
v2'
v3
v3'v4
v4'
v5v5'
v6
v6'v7
v7'
u1'
u1
u2u2'
u3
u3'
u4
u4'
u5
u5'
u6' u6
u7
u7'
v0'6
4
12
10
14
1
3
5
11
7
13
2
17
15
19
18
16
20 8
923
22
21
24
27
2526
28
29
30
Figure 6
Definition 2.4 ([4]). The fan Fn is defined as the join of Pn and K1. The vertex corresponding to
K1 is said to be the apex vertex. The fan Fn is shell Sn+1.
Theorem 2.7. Fn �K1 is a sum divisor cordial graph.
Proof. Let v0, v1, v2, . . . , vn be the vertices of the fan Fn, where v0 be apex vertex.
Let v′0, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph Fn �K1.
V (Fn �K1) = V (Fn) ∪ {vi, ; 0 ≤ i ≤ n}.10
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E(Fn �K1) = E(Fn) ∪ {viv′i; 0 ≤ i ≤ n}.|V (Fn �K1)| = 2n + 2.
|E(Fn �K1)| = 3n.
We define labeling f : V (Fn �K1)→ {1, 2, 3, . . . , 2n + 2} as follows.
For n ≡ 0, 3(mod4) :
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 1;
f(v′0) = 2n + 2;
f(v′i) = (n + 1) + i 1 ≤ i ≤ n.
For n ≡ 1(mod4) :
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 1;
f(v′0) = 2n + 1;
f(v′i) = (n + 1) + i 1 ≤ i ≤ n− 1;
f(v′n) = 2n + 2.
For n ≡ 2(mod4) :
f(vi) =
i + 1 i ≡ 0, 1(mod 4);
i + 2 i ≡ 2(mod 4);
i i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v0) = 1;
f(v′0) = 2n + 1;
f(v′i) = n + 1 + i 2 ≤ i ≤ n− 1;
f(v′1) = n + 1;
f(v′n) = 2n + 2.
In view of above defined labeling pattern we have the following.
Cases of n Edge conditions
n ≡ 1(mod 4) ef (1) =⌈3n2
⌋, ef (0) =
⌊3n2
⌉
n ≡ 3(mod 4) ef (1) =⌊3n2
⌋, ef (0) =
⌈3n2
⌉
n ≡ 0, 2(mod 4) ef (1) = 3n2 = ef (0)
Thus |ef (0)− ef (1)| ≤ 1.
Hence the graph under consideration admits sum divisor cordial labeling.
Hence Fn �K1 is a sum divisor cordial graph.
Example 2.7. Sum divisor cordial labeling of F8 � K1 is shown in Figure 7 as an illustration for
Theorem 2.7.
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v1 v2 v4 v5
v2'v1' v3' v4' v5'
u1'
u1
v6 v7
v6' v7'
v3 v8
v8'
64 8
1210 14
1
3 5
11
97
13
2
1715
18
16
Figure 7
Definition 2.5 ([4]). The Double fan DFn is defined as the join Pn + 2K1.
Theorem 2.8. DFn �K1 is a sum divisor cordial graph.
Proof. Let u0, v0, v1, v2, . . . , vn be the vertices of the double fan DFn, where u0 and v0 are the vertices
of degree n and v1, v2, . . . , vn are the vertices corresponding to path Pn.
Let u′0, v′0, v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph DFn �K1.
V (DFn �K1) = V (DFn) ∪ {u′0} ∪ {v′i, ; 0 ≤ i ≤ n}.E(DFn �K1) = E(DFn) ∪ {u0u
′0} ∪ {viv′i; 0 ≤ i ≤ n}.
|V (DFn �K1)| = 2n + 4.
|E(DFn �K1)| = 4n + 1.
We define labeling f : V (DFn �K1)→ {1, 2, 3, . . . , 2n + 4} as follows.
For n ≡ 0, 2(mod4)
f(u0) = 1;
f(u′0) = 2n + 4;
f(v0) = 2;
f(v′0) = 2n + 3;
f(vi) = 2 + i 1 ≤ i ≤ n;
f(v′i) = n + 2 + i 1 ≤ i ≤ n.
For n ≡ 1, 3(mod4)
f(u0) = 1;
f(u′0) = 2n + 2;
f(v0) = 2;
f(v′0) = 2n + 4;
f(vi) = 2 + i 1 ≤ i ≤ n;
f(v′2i−1) = n + (2i− 1) + 3 1 ≤ i ≤ n + 1
2;
f(v′2i) = n + 2i + 1 1 ≤ i ≤ n− 1
2.
In view of above defined labeling pattern we have the following.
Cases of n Edge conditions
n ≡ 1, 3(mod 4) ef (1) =⌈4n+1
2
⌉, ef (0) =
⌊4n+1
2
⌋
n ≡ 0, 2(mod 4) ef (1) =⌊4n+1
2
⌋, ef (0) =
⌈4n+1
2
⌉
12
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Thus |ef (0)− ef (1)| ≤ 1.
Hence the graph under consideration admits sum divisor cordial labeling.
Hence DFn �K1 is a sum divisor cordial graph.
Example 2.8. Sum divisor cordial labeling of DF6 �K1 is shown in Figure 8 as an illustration for
Theorem 2.8.
v4 v5
v2'v1' v3' v4' v5'
u1'
u2
u2'
u1
v6
v6'
v2 v3
3 4 5 6 7 8
1
2
15
16
9 10 11 12 13 14
v1
Figure 8
Definition 2.6 ([4]). Let G = (V,E) be a graph. If every edge of graph G is subdivided, then the
resulting graph is called barycentric subdivision of graph G.
In other words, barycentric subdivision is the graph obtained by inserting a vertex of degree 2 into
every edge of original graph.
The barycentric subdivision of any graph G is denoted by S(G).
Theorem 2.9. S(K1,n)�Kn is a sum divisor cordial graph.
Proof. Let v0, v1, v2, . . . , vn, w1, w2, . . . , wn be the vertices of graph S(K1,n), where v0 be apex vertex,
v1, v2, . . . , vn be the vertices of degree 1 and w1, w2, . . . , wn be the vertices of degree 2.
Let v′1, v′2, . . . , v
′n, w′1, w
′2, . . . , w
′n be the newly added vertices to obtain the graph S(K1,n)�K1.
V (S(K1,n)�K1) = V (S(K1,n)) ∪ {v′0, v′i, w′i; 1 ≤ i ≤ n}.E(S(K1,n)�K1) = E(S(K1,n) ∪ {v0v′0, viv′i, wiw
′i; 1 ≤ i ≤ n}.
|V (S(K1,n)�K1)| = 4n + 2.
|E(S(K1,n)�K1)| = 4n + 1.
We define labeling f : V (S(K1,n)�K1)→ {1, 2, 3, . . . , 4n + 2} as follows.
f(v0) = 1;
f(v′0) = 4n + 2;
f(wi) = 2i + 1 1 ≤ i ≤ n;
f(w′i) = 2i 1 ≤ i ≤ n;
f(vi) = (2n + 1) + 2i 1 ≤ i ≤ n;
f(v′i) = 2n + 2i 1 ≤ i ≤ n.
In view of above defined labeling pattern we have the following.
ef (1) =
⌊4n + 1
2
⌋,
ef (0) =
⌈4n + 1
2
⌉.
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Thus |ef (0)− ef (1)| ≤ 1.
Hence the graph under consideration admits sum divisor cordial labeling.
Hence S(K1,n)�K1 is a sum divisor cordial graph.
Example 2.9. Sum divisor cordial labeling of S(K1,5)�K1,5 is shown in Figure 9 as an illustration
for Theorem 2.9.
u1
v0 v1 v1'
v2
v2'
v3
v3'
v4
v4'v5
v5'
u2
u2'
u1'
u3
u4
u4'
u5
u3'
u5'
v0'
1
3 2
4
7
6
8
9
10
11
12
14
13
15
5
16
17
18
19
20
22
21
Figure 9
Definition 2.7 ([4]). A chord of a cycle Cn is an edge joining two non-adjacent vertices of cycle Cn.
Theorem 2.10. G�K1 is a sum divisor cordial graph, where G is cycle with one chord and chord
forms a triangle with two edges of Cn.
Proof. Let G be the cycle Cn with one chord.
Let v1, v2, . . . , vn be vertices of Cn and e = v2vn be the chord of Cn.
Let v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph G�K1.
V (G�K1) = V (Cn) ∪ {v′i; 1 ≤ i ≤ n},E(G�K1) = E(Cn) ∪ {viv′i; 1 ≤ i ≤ n}.Hence |V (G�K1)| = 2n and |E(G�K1)| = 2n + 1.
We define labeling f : V (G�K1)→ {1, 2, 3, . . . , 2n} as follows.
f(vi) = 2i− 1 1 ≤ i ≤ n;
f(v′i) = 2i 1 ≤ i ≤ n.
In view of above defined labeling pattern we have the following.
ef (1) =
⌈2n + 1
2
⌉,
ef (0) =
⌊2n + 1
2
⌋.
Thus |ef (0)− ef (1)| ≤ 1.
Hence G�K1 is a sum divisor cordial graph.
Example 2.10. Sum divisor cordial labeling of corona of C6 with one chord and K1 is shown in
Figure 10 as an illustration for Theorem 10.
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v1
v2
v3
v4
v5
v6
v1'
v2'
v3'
v4'
v5'
v6'
1
3
4
6
7
8
5
2
9
10
11
12
Figure 10
Definition 2.8 ([4]). Two chords of a cycle are said to be twin chords if they form a triangle with
an edge of the cycle Cn.
For positive integers n and p with 3 ≤ p ≤ n−2, Cn,p denotes the graph consisting of a cycle Cn with
twin chords with which the edges of Cn form cycles Cp, C3 and Cn+1−p without chords.
Theorem 2.11. Cn,3 �K1 is a sum divisor cordial graph.
Proof. Let v1, v2, . . . , vn be the vertices of Cn, e1 = v2vn and e2 = v3vn be the chords of Cn.
Let v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph Cn,3 �K1.
V (Cn,3 �K1) = V (Cn,3) ∪ {v′i; 1 ≤ i ≤ n}.E(Cn,3 �K1) = E(Cn,3) ∪ {viv′i; 1 ≤ i ≤ n}.|V (Cn,3 �K1)| = 2n.
|E(Cn,3 �K1)| = 2n + 2.
We define labeling f : V (Cn,3 �K1)→ {1, 2, 3, . . . , 2n} as follows.
For n ≡ 0, 1, 3(mod4) :
f(vi) =
i i ≡ 0, 1(mod 4);
i + 1 i ≡ 2(mod 4);
i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′i) = n + 1 1 ≤ i ≤ n.
For n ≡ 2(mod4) :
f(vi) =
i i ≡ 0, 1(mod 4);
i + 1 i ≡ 2(mod 4);
i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′i) = n + i 3 ≤ i ≤ n;
f(v′1) = n;
f(v′2) = n + 2.
In view of above defined labeling pattern we have the following.
ef (1) = n + 1 = ef (0)
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Thus |ef (0)− ef (1)| ≤ 1.
Hence Cn,3 �K1 is a sum divisor cordial graph.
Example 2.11. Sum divisor cordial labeling of C7,3 �K1 is shown in Figure 11 as an illustration
for Theorem 11.
v1
v1'
v2
v2'
v3 v3'
v4
v4'
v5
v5'
v6v6'
v7
v7' 1
3
2
45
7
6
8
9
10
1112
14
13
Figure 11
Definition 2.9 ([4]). A cycle with triangle is a cycle with three chords which by themselves form a
triangle.
For positive integers p, q, r and n ≥ 6 with p+ q + r + 3 = n, Cn(p, q, r) denotes a cycle with triangle
whose edges form the edges of cycles Cp+2, Cq+2 and Cr+2 without chords.
Theorem 2.12. Cn(1, 1, n− 5)�K1 is a sum divisor cordial graph.
Proof. Let v1, v2, . . . , vn be the vertices of Cn.
Let e1 = u1u3, e2 = u3un−1 and e3 = u1un−1 be chords of Cn which by themselves form a triangle.
Let v′1, v′2, . . . , v
′n be the newly added vertices to obtain the graph Cn(1, 1, n− 5)�K1.
V (Cn(1, 1, n− 5)�K1) = V (Cn) ∪ {v′i; 1 ≤ i ≤ n},E(Cn(1, 1, n− 5)�K1) = E(Cn(1, 1, n− 5)) ∪ {viv′i; 1 ≤ i ≤ n.}Hence |V (Cn(1, 1, n− 5)�K1)| = 2n and |E(Cn(1, 1, n− 5)�K1)| = 2n + 3.
We define labeling f : V (Cn(1, 1, n− 5)�K1)→ {1, 2, 3, . . . , 2n} as follows.
For n ≡ 0, 1, 3(mod4) :
f(vi) =
i i ≡ 0, 1(mod 4);
i + 1 i ≡ 2(mod 4);
i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′i) = n + i 1 ≤ i ≤ n.
For n ≡ 2(mod4) :
f(vi) =
i i ≡ 0, 1(mod 4);
i + 1 i ≡ 2(mod 4);
i− 1 i ≡ 3(mod 4); 1 ≤ i ≤ n.
f(v′i) = n + i 3 ≤ i ≤ n;
f(v′1) = n;
f(v′2) = n + 2.
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In view of above defined labeling pattern we have the following.
ef (1) =
⌊2n + 3
2
⌋,
ef (0) =
⌈2n + 3
2
⌉.
Thus |ef (0)− ef (1)| ≤ 1.
Hence Cn(1, 1, n− 5)�K1 is a sum divisor cordial graph.
Example 2.12. Sum divisor cordial labeling of C8(1, 1, 3)�K1 is shown in Figure 12 as an illustration
for Theorem 12.
v1
v1'
v2
v2'
v3 v3'
v4
v4'v5
v5'
v6
v6'
v7
v8
v7'
1
3
2
4
5
7
6
8
9
10
11
12
13
16
15
14
Figure 12
3 Concluding Remarks
The sum divisor cordial labeling is an invariant of divisor cordial labeling by considering codomain
as finite set of numbers. It is interesting to see that if two graphs are sum divisor cordial then their
corona is sum divisor cordial or not. We have investigated twelve sum divisor cordial graphs in context
of corona of graphs.
References
[1] D. G. Adalja and G. V. Ghodasara, Some New Sum Divisor Cordial Graphs, International
Journal of Applied Graph Theory, Vol. 2, No. 1, 2018, pp. 19 - 33.
[2] D. M. Burton, Elementary Number Theory, Brown Publishers, Second Edition, (1990).
[3] R. Frucht and F. Harary, On Corona of Two Graphs, Aequationes Mathematicae, Vol. 4, No. 3,
1970. DOI: 10.1007/BF01817769
[4] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics,
20(2017), # DS6, pp. 1 - 432.
[5] G. V. Ghodasara and D. G. Adalja, Divisor Cordial Labeling in Context of Ring Sum of Graphs,
International Journal of Mathematics and Soft Computing, Vol. 7, No. 1, 2017, pp. 23 - 31.
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[6] J. Gross and J. Yellen, Graph Theory and Its Applications, CRC Press, (2004).
[7] A. Lourdusamy and F. Patrick, J. Shiama, Sum Divisor Cordial Graphs, Proyecciones Journal
of Mathematics, Vol. 35, No. 1, 2016, pp. 119 - 136.
[8] A. Lourdusamy and F. Patrick, Sum Divisor Cordial Labeling For Star And Ladder Related
Graphs, Proyecciones Journal of Mathematics, Vol. 35, No. 4, 2016, pp. 437 - 455.
[9] A. Rosa, On certain valuations of the vertices of theory of graphs, (Internat.Symposium, Rome,
July 1966) Gordon and Breach, N. Y. and Dunod Paris (1967), pp. 349 - 355.
[10] R. Varatharajan and S. Navanaeethakrishnan and K. Nagarajan, Divisor Cordial Graphs, Inter-
national J. Math. Combin., Vol. 4, 2011, pp. 15 - 25.
[11] V. Yegnanaryanan and P. Vaidhyanathan, Some Interesting Applications of Graph Labellings,
J. Math. Comput. Sci., Vol. 2, No.5, 2012, pp.1522 - 1531.
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