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INTRODUCTIONshodhganga.inflibnet.ac.in/bitstream/10603/97325/6/06... · 2018-07-08 · consecutive...
Transcript of INTRODUCTIONshodhganga.inflibnet.ac.in/bitstream/10603/97325/6/06... · 2018-07-08 · consecutive...
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INTRODUCTION
One of the most interesting problems in the area of Graph
Theory is that of labeling of graphs. A labeling or valuation or numbering
of a simple graph G is an one-to-one mapping from its vertex set into a
set of non-negative integers which induces an assignment of labels to the
edges of G.
Labeled graphs serve as useful models for a broad range of
applications. They are useful in many coding theory problems, including
the design of good radar type codes [25], Synch - set codes, missile
guidance codes and convolutional codes with optimal non-standard
encodings of integers. Labeled graphs have also been applied in
determining ambiguities in X-ray crystallographic analysis, and designing
a communication network addressing system in determining optional
circuit layouts and radio astronomy problems.
The impatus for graph labeling problem began with the
conjecture of G. Ringel [101] which states that "All trees are graceful".
In 1967 Rosa [99] defined ~-valuations of G and Golomb
[59] called such labelings graceful. Much work was done with graceful
labeling. A large number of families has been numbered gracefully. See
{ [1], [2], [3], [5], [10], [13], [21], [25], [26], [27], [28], [32], [33], [34], [38],
[42], [44], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [58],
[59], [66], [69], [72], [74], [76], [77], [78], [79], [80], [82], [88], [89], [92],
[93], [96], [98], [99], [100], [101], [104], [107], [108], [109], [110], [111],
[112], [115], [116], [117], [118]}.
Apart from the graceful labeling, some other labelings were
defined and developed. Graham and Sloane [61] introduced harmonious
graphs {see [24], [41], [54], [55], [56], [57], [58], [61], [90], [102], [103] }.
Some of the other known labelings are sequential, Arithmetic { [4], [6], [8],
[23], [37], [60], [67], [71], [108], [109], [114], [115], [119], [120], [121] }
indexable labeling { [9], [12], [22] }. Edge graceful labeling { [91], [94],
[95] } Sum labeling { [11], [35], [64] }, Skolem graceful { [16], [43], [87] },
Felicitous labeling { [22], [86], [106] }, set sequential labeling [7], Magic
labeling { [15], [17], [18], [19], [20] }
The thesis, II A Study on different classes of graphs and
their labelings", proposes a study of different types of labelings and
various classes of graphs that possess such labelings.
This thesis consists of five chapters. Special emphasis has
been given to Triangular gracefulgraphs, Antimagic graphs, Consecutive
graphs and E-cordial graphs.
In chapter I, we give a brief survey of graph labelings and
we collect some basic definitions and results which are useful for the
subsequent chapters. For basic graph theoretic terminologies,
definitions and notations we follow { [31], [63], [97] }.
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Chapter II concentrates on Triangular graceful graphs. We
.[48] introduced triangular graceful graphs. In Number Theory [113]
triangular numbers are defined.
In this chapter we prove the following graphs, are triangular
graceful. Path Pm, olive trees, caterpillars, the star Sk,m, the one point
union of k copies of path of length m, cycles Cn, n == 0 (mod 4), We also
prove certain classes of graphs such as wheels, Kn, Km n, m,n ;t:. 1, are,
not triangular graceful. We set forth the conjecture that all trees are
triangular graceful.
Chapter III deals with Antimagic graphs. In this chapter we
prove the following graphs are Antimagic.
1. Crown Cn 0 K1
2. Cn 0 Pm [for m =nand m =n + 1 ]
3. The double crown Cn 0 P2 \;j odd n ~ 3
4. < Cn, t > I the one point union of t copies of Cn
5. The grid Cn x Cn for even n
6. Pn + K t
7. Pn X C3 for all even n
8. Triangular snakes, quadrilateral snakes, K4 snakes
9. K1 n + K t
10. Book graph Bn
In Chapter IV, we study the consecutive labeling. In this
chapter we prove the following graphs are consecutive graphs.
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1. The prism Cn X P2 has magic vertex labeling, and edge
consecutive labeling.
2. The n-gonal snakes are edge consecutive.
3. Quadrilateral and triangular snakes are edge consecutive
4. The graph Cn 11 Cm is edge consecutive.
5. < Cn m >, the one point union of m copies of cycle Cn, has,
consecutive labeling.
Also we construct some classes of graphs and prove they
are consecutive.
In chapter V we consider E-cordial graphs. In this chapter
we prove the following graphs are E-cordial.
1. K2 0 Cn, V n :f. 0 (mod 2)
2. Cn 0 P3 V even n
3. Closed helm W*(t,n) for t = 2,4,6
4. The graph L1Cn, Lotus inside a circle Vn,
5. Book graph Bn when n is odd
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6. Mirror sum graph M(m,n) when m + n == 0 (mod 2)
7. The generalised Petersen graph P(n,k) for all k and even n.
8. The graph Hn,n with vertex set {u1,u2, ... ,un,v1,v2, ... ,vn} and the edge
set {UjVj 11 ~ i ~ j ~ n } is E-cordial for n == 0 (mod 4)
10. Ladder Ln = Pn X K2
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