CHAPTER III SUM OF PATH AND NULL GRAPH -...

68

CHAPTER III

SUM OF PATH AND NULL GRAPH

Introduction: A number of classes of graphs that are the join of graphs

have been shown that to be graceful. Acharya [1982] proved that if G is a

connected graph, then G + Kn is graceful. Bhat-Nayak and Gohkale

[1986] found that Kn +2K2 is not graceful whereas Amutha and

Kathiresan got that the graph obtained by attaching a pendant edge to

each vertex of Kn + 2K2

is graceful.

Lee and Wui [1987] obtained that the disjoint union of 2 or 3 stars

is Skolem - graceful if and only if at least one star has even size. Lee,

Wang, and Wui [1988] showed that the 4-star St(n1, n2, n3, n4

) is

Skolem - graceful for some special cases and conjectured that all 4-stars

are Skolem - graceful. Denham, Leu, and Liu [1993] proved this

conjecture.

Gnanajothi [1991] determined the line-graceful graphs are: Pn if

and only if n ≡ 2 (mod 4); C n if and only if n ≡ 2 (mod 4); K 1,n if and

only if n ≡ 1 (mod 4); Pn K1(combs) if and only if n is even; (Pn K1)

K1 if and only if n ≡ 2 (mod 4); (in general, if G has order n, G H is

the graph obtained to every vertex in the ith copy of H); mCn when mn is

odd; Cn K1 (crowns) if and only if n is even; mC4 for all m; complete

n-ary trees when n is even; K1,n ∪ K1,n if and only if n is odd; odd cycles

with a chord; even cycles with a tail; even cycles with a tail of length 1

and a chord;

69

Section 3.1- Preliminaries and Previous works: Graphs consisting of two triangles having a common vertex and

tails of equal length attached to a vertex other than the common one; the complete n-ary tree when n is even; trees for which exactly one vertex has even degree. She conjectured that all trees with p ≡ 2 (mod 4) vertices are line - graceful and proved this conjecture for p ≤ 9.

Gnanajothi [1991] investigated the line - gracefulness of several graphs obtained from stars. In particular, the graph obtained from K1,4 by subdividing one spoke to form a path of even order (counting the centre of the star) is line-graceful; the graph obtained from a star by inserting one vertex in a single spoke is line-graceful if and only if the star has p ≡ 2 (mod 4) vertices; the graph obtained from K1,n

by replacing each spoke with a path of length m (counting the centre vertex) is line - graceful in the following cases: n = 2; n = 3 and m ≡ 3 (mod 4); and m is even and mn +1 ≡ 0 (mod 4).

She also studied graphs obtained by joining disjoint graphs G and H with an edge. She proved such graphs are line - graceful in the following circumstances: G = H; G = Pn, H = Pm and m + n ≡ 0 (mod 4); and G = Pn K1, H = Pm K1 and m + n ≡ 0 (mod 4). She called a graph G bigraceful if both G and its line graph are graceful. She showed the following are bigraceful: Pm; Pm × Pn; Cn if and only if n ≡ 0, 3 (mod 4); S n; Kn if and only if n ≤ 3; and B n

if and only if n ≡ 3(mod 4).

Lee, Wang, and Kiang [1994] discussed that P (P2k, f ) a graceful

when f = (12), (13),….,(k, k+1),….,(2k-1, 2k). They conjectured that if

G is a graceful on bipartite graph with n vertices, then for any

permutation f on 1, 2, . . , n, the permutation graph P (G, f ) is graceful.

Balakrishnan and Sampathkumar [1996] asked for which m ≥ 3 is the

graph Kn + mK2 graceful for all n.

70

Choudum and Kishore [1996] analysed that the disjoint union of k

copies of the star K1,2p is Skolem graceful if k ≤ 4p + 1 and the disjoint

union of any number of copies of K1,2 is Skolem graceful. They also

proved that all 5-stars are Skolem graceful. Kishore [1996] has shown

that a necessary condition for St(n1, n2, . . . , nk) to be Skolem graceful is

that some ni

is even or k ≡ 0 or 1 (mod 4). He conjectures that each one

of these conditions is sufficient.

Ramirez- Alfosin [1999] obtained that if G is graceful and

│V(G) │= │E(G)│= e and either 1 or e is not a vertex label then G + Kt

is graceful for all t. He also proved that Rm(Pn) is graceful for all m and

all n > 1 and that R(m,1,..,1) (C4n), R(2,1,..,1) (Cn) (n ≥ 8) and R (2,2,1,..,1) (C4n)

(n ≥ 8) are graceful. Sethuraman and Kishore [1999] determined the

graceful graphs that are the union of n copies of K4

with i edges deleted

for 1 ≤ i ≤ 5 with one edge in common.

Kathiresan [2000] used the notation Pa,b to denote the graph

obtained by identifying the end points of b internally disjoint paths each

of length a. He conjectured that Pa,b

is graceful except when a is odd and

b ≡ 2 (mod 4). He proved the conjecture for the case that a is even and b

is odd.

Sethuraman and Elumalai [2001] got that K1,m,n with a pendant

edge attached to each vertex is graceful. Sekar [2002] found that Pa,b is

graceful when a = 4r + 1, r > 1; b = 4m, m > r. Wu [2002] proved that if

G is a graceful graph with n edges and n + 1 vertices then the join of G

and Km

and the join of G and any star are graceful.

71

Redl [2003] discussed that the double cone Cn + K2 is graceful for

n = 3, 4, 5, 7, 8, 9,11 and 12 but not graceful for n≡ 2 (mod 4). Yousef

[2003] has shown that Kn+ mK2 is graceful if m ≡ 0 or 1 (mod 4) and

that Kn+ mK2 is not graceful if n is odd and m ≡ 2 or 3 (mod 4). Pan and

Lu [2003] have shown that (P2 + Kn) ∪ K1,m and (P2 + Kn) ∪ Tn

are

graceful.

Jirimutu [2003] investigated that the graph obtained by attaching a

pendant edge to every vertex of Km,n is graceful. The n-cone (also called

the n-point suspension of Cm) Cm + Kn has been shown to be graceful

when m ≡ 0 or 3 (mod 12) by Bhat-Nayak and Selvam [2003]. When n is

even and m is 2, and Selvam [2003] also proved that the following cones

are graceful: C4 + Kn, C5 + K2, C7 +Kn, C9 + K2, C11 + Kn and C19 + Kn

.

For i = 1, 2, . . . , m let vi,1, vi,2, vi,3, vi,4 be a 4-cycle. Yang and

Pan [2003] defined F k,4 to be the graph obtained by identifying vi,3 and

vi+1,1 for i = 1, 2, . . ,(k – 1). They proved that Fm1 ,4 ∪ Fm2 ,4 ∪ · · · ∪ Fmn ,4

is graceful for all n. Lee, Chen, and Wang [2004] have determined the

edge - graceful spectra for various cases of cycles with a chord and for

certain cases of graphs obtained by joining two disjoint cycles with an

edge (i.e., dumbbell graphs).

Yang, Rong, and Xu [2004] determined that Pa,b is graceful when

a = 10, 12, and 14 and b is even. They also proved that Pa,b is graceful

when a = 3, 5, 7, and 9 and b is even and when a = 2, 4, 6, and 8 and b is

even. Kathiresan also shown that the graph obtained by identifying a

vertex of Kn with any non centre vertex of the star with 2n-1-n(n-1)/2

edges is graceful.

72

Lee, Chen, Yera, and Wang [2004] proved that if G is a super

edge - graceful with p vertices and q edges and q ≡ −1 (mod p) when q is

even, or q ≡ 0 (mod p) when q is odd, then G is also edge - graceful.

They also obtained: the graph obtained from connected super

edge - graceful unicyclic graph of even order by joining any two

nonadjacent vertices by an edge is super edge-graceful; the graph

obtained from a super edge - graceful graph with p vertices and (p + 1)

edges by appending two edges to any vertex is super edge - graceful; and

the one - point union of two identical cycles is super edge - graceful. Barrientos [2005] also discussed: if G is a graceful graph of order

m and size m−1, G nK1 and G + nK1 are graceful; if G is a graceful

graph of order p and size q with q > p, then (G ∪ (q + 1 − p)K1) nK1

is

graceful; and all unicyclic caterpillar are graceful.

Hefetz [2005] investigated that a graph G = (V, E) of the form

H ∪ f1 ∪ f2 ∪ …∪ fr where H = (V, E ) is edge - graceful and the fi’s

are 2-factors is also edge-graceful and that a regular graph of even degree

that has a 2-factor consisting of k cycles each of length t where k and t

are odd is edge - graceful. Youssef [2006] found that for all n ≥ 2,

Pn ∪ Sm is Skolem - graceful if and only if n ≥ 3 or n = 2 and m is even.

He also got that if G is Skolem - graceful, then G + Kn

is graceful.

Shiu, Ling, and Low [2006] obtained the entire edge - graceful spectra of

cycles with one chord.

In this chapter the following connected graphs (P2 + Nn)°e,

P3 + Nn , P4 + Nn , P5 + Nn, K1 + Pn, K1 + 2Pn, K1 + 3Pn, K1 + 4Pn and

K1 + 5Pn

are proved as edge - odd graceful.

73

Section 3.2- Edge - odd graceful labeling of (P2+Nn

Definition 3.2.1: (P

)°e:

2+Nn)°e is (P2+Nn) merging by an edge and it is a

connected graph such that every vertex of (P2+Nn) is adjacent to every

vertex of null graph Nn together with adjacent edges in (P2+Nn

).

Theorem 3.2.2: The connected graph (P2+Nn

Proof: The graph (P

)°e is edge - odd graceful.

2+Nn)°e is a connected graph with (n+2) vertices

and (2n+2) edges and the arbitrary labelings for vertices and edges for

(P2+Nn)°e

are mentioned below:

V2 e1 V1

e

2

e3e4e5 e6e7 e8 …. en en+1 e

n+2

en+3 en+4 en+5 en+6 en+7 en+8 en+9 e2n-1 e2n e2n+1 e

…….. 2n+2

V3 V4 V5 V6 V2n+1 V2n+2

Figure 3.01: Edge - odd graceful labeling of (P2+Nn

)°e

To find edge - odd graceful, define f: E((P2+Nn

f(e

)°e ) → {1, 3, …, 2q-1}

by

i

) = 2i-1,i = 1, 2, 3,…,(2n+2) Rule(1)

Define f+: V(G) → {0, 1, 2, …, (2k -1)} by f+

(v) ≡ Σ f(uv) mod (2k),

where this sum run over all edges through v …Rule (2)

74

Hence the induced map f+ provides the distinct labels for vertices

and also the edge labeling is distinct. So the connected graph (P2+Nn

)°e

is edge - odd graceful.

Example 3.2.3: The connected graph (P2+N6

Proof: The graph (P

)°e is edge - odd graceful.

2 + N6) °e

is a connected graph with 8 vertices and

14 edges. Due to the rules (1) & (2) in theorem (3.2.2), edge - odd

graceful labelings of the required graph is obtained as follows: 1

8 3 24

5 7 9 11 13 15 17 19 21 23 25 27

22 26 2 6 10 14

Figure 3.02: Edge - odd graceful labeling of (P2+N6

)°e

Section 3.3- Edge - odd gracefulness of P3 +Nn:

Definition 3.3.1: Pk + Nn is a connected graph such that every vertex of

Pk is adjacent to every vertex of null graph Nn together with adjacent

edges in Pk

.

Lemma 3.3.2: The connected graph P3 +N2

Proof: The graph P


3 +N2

is a connected graph with 5 vertices and 8

edges. The arbitrary labeling of edge - odd graceful of the required graph

is obtained as follows:

3 3 8 1 9

9

5 11 13 15

7

75

Figure 3.03: Edge - odd graceful labeling of P

5 7

3 +N

2

Theorem 3.3.3: The connected graph P3+Nn

Proof: Let {v

is edge-odd graceful.

i:1 ≤ i ≤ n+3} be the vertices {e i:1 ≤ i ≤ 3n+2} be the

edges of the graph P3+Nn. It is a connected graph such that every vertex

of P3 is adjacent to every vertex of null graph Nn together with adjacent

edges in P3. The arbitrary labelings for vertices and edges for P3 +Nn

are

mentioned below:

v3 e2 v2 e1 v

1

…en+1 en+2 en+3en+4….. e2n e2n+1 e

e2n+2

3 e4 e5 en+5 e

n+6

e2n+3 e2n+4 e2n+5 e2n+6 …….. e3n e3n+1 e

…………… 3n+2

v4 v5 v6 v7 vn+1 vn+2 v

n+3

Figure 3.04: Edge - odd graceful labeling of P3 +N

n

For n ≡ 6 (mod 8), the arbitrary labelings for vertices and edges for

P3 +Nn


v3 e3n+2 v2 e3n+1 v

1

… en-1 en en+1en+2 ...e2n-1e2n

e

1 e2 e3 en+3

e2n+1 e2n+2 e2n+3 e2n+4 ……..….. e3n-1 e

3n

v4 v5 v6 v7……………. vn+1 vn+2 vn+3

76


n

To find edge - odd graceful, define f: E(P3 +Nn ) → {1, 3, …, 2q-1} by

Case i: n ≡ 1(mod 6)

n is odd

f(e1) = 5, f(e3

f(e

) = 1

i

) = 2i-1, i = 2, 4, 5, 6,…,(3n+2) Rule (1)

Case ii: n ≡ 3, 5 (mod 6)

f(ei) = 2i-1, i = 1, 2, 3,…,(3n+2) Rule (2)

Case iii: n ≡ 0 (mod 8)

n is even

f(e1) = 5, f(e3

f(e

) = 1

i

) = 2i-1, i = 2, 4, 5, 6,…,(3n+2) Rule (3)

Case iv: n ≡ 2 (mod 8)

f(e1) = 9, f(e5

f(e

) = 1

i

) = 2i-1, i = 2, 3, 4, 6,…,(3n+2) Rule (4)

Case v: n ≡ 4 (mod 8)

f(en) = 4n-1, f(e2n

f(e

) = 2n-1

i

(2n+1), …, (3n+2) Rule (5)

)=2i-1,i=1, 2,…,(n-1), (n+1),…,(2n-1),

Case vi: n ≡ 6 (mod 8)

f(e1) =7, f(e3) = 1, f(e4

f(e

) = 5

i

) = 2i-1, i = 2, 5, 6,…,(3n+2) Rule (6)

77

Define f+: V(G) → {0, 1, 2, …, (2k-1)} by f+

(v) ≡ Σ f(uv) mod (2k),

where this sum run over all edges through v …. Rule (7)


and also the edge labeling is distinct. Thus the connected graph P3 + Nn


Example 3.3.4: The connected graph P3 +N7

Proof: The graph P


3 +N7


edges, where n ≡ 1 (mod 6). Due to the rules (1) & (7) in theorem (3.3.3),

edge - odd graceful labeling of the required graph is obtained as follows:

30

3 45 5 12

1 7 9 11 13 15 17

19 21 23 25 27 29 31 33 35 37 39 41 43 45

7 17 23 29 35 41 1


7


Proof: The graph P


3 +N3 is a connected graph with 6 vertices and 11


edge - odd graceful labeling of the required graph is obtained as follows: 2

3 21 1 14

11 13 15 17 19 21

5 7 9

11 17 1

78


Example 3.3.6: The connected graph P3

3 +N5

Proof: The graph P


3 +N5



edge - odd graceful labelings of the required graph is obtained as follows:

14

3 31 1 10

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

11 17 23 29 1


5


Proof: The graph P


3+ N8




43

3 24 5 45

21 23 252729 3133 35 37 39 41 43 45 4749 51

1 7 9 11 13 15 17 19

7 17 23 29 35 41 47 1

Figure 3.09: Edge - odd graceful labeling of P3 +N8

79


Proof: The graph P


3 +N10




7

3 32 9 37

5 7 1 11 13 15 17 1921 23 25 2729 31 33 35 37 39 41 43

45 47 49 51 53 55 57 59 61 63

11 17 15 29 35 41 47 53 59 1


10


Proof: The graph P


3+ N4




23 27 8 25 21

1 3 5 15 9 11 13 7 17 19 21 23

27 5 11 17

80


Example 3.3.10: The connected graph P4

3 +N6

Proof: The graph P


3 + N6



edge - odd graceful labelings of the required graph is obtained as follows:

17

3 22 7 11

1 5 9 11 13 15

17 19 21 23 25 27 29 31 33 35 37 39

Figure 3.12: Edge - odd graceful labeling of P 7 15 23 29 35 1

3 +N

6

Section 3.4 - Edge - odd gracefulness of P4 + Nn :

Theorem 3.4.1: The connected graph P4 + Nn

Proof: Let {v


i: 1 ≤ i ≤ (n + 4)} be the vertices {ei: 1 ≤ i ≤ (4n+3)} be

the edges of the graph P4+Nn. It is a connected graph such that every

vertex of P4 is adjacent to every vertex of null graph Nn together with

adjacent edges in P4. The arbitrary labelings for vertices and edges

for P4 + Nn

vare mentioned below:

4 e3 v3 e2 v2 e1 v

1

…en+2 en+3 en+6en+7…. e2n+2 e

e2n+3

4 e5 e6 en+4e

en+5

2n+4 e2n+5e2n+6 …… 1 e3n+2 e3n+3

e3n+4 e3n+5 e3n+6 …….. e4n+1 e4n+2 e

4n+3

v5 v6 v7 v8 v9……. vn+2 vn+3

vn+4

81


n

n ≡ 5 (mod 6) v4 e4n+3 v3 e4n+2 v2 e4n+1 v

1

…e3n+2 e3n+1 e2n-2e2n-3 en+2 e

en+1

4n e4n-1 e4n-2 e2ne

e2n-1

n en-1en-2 …… e3 e2 e1

e3n e3n-1 e3n-2 …… e2n+3 e2n+2 e

2n+1

v5 v6 v7 v8 v9 …… vn+2 vn+3 v

n+4


n

To find edge - odd graceful, define f: E(P4 +Nn ) → {1, 3, …, 2q-1} by

Case i. n ≡ 1(mod 6)

n is odd

f(ei

f(e

)=2i-1, i=1,2,…,(n+2), (n+4),…,(4n+2) Rule (1)

n+3) = 8n+5, f(e4n+3

Case ii. n ≡ 3 (mod 6)

) = 2n+5

f(e1) =5, f(e2) =1, f(e3

f(e

) =3

i

)=2i-1, i = 4,5, 6,…,(4n+3) Rule (2)

Case iii. n ≡ 5 (mod 6) f(e1) = 4n+1, f(e2n+1

f(e

) = 1

i) = 2i-1,i = 2,3,…,(2n), (2n+2),…,(4n+3) Rule (3)

Case iv. n ≡ 0, 2 (mod 6)

n is even

82

f(ei

) = 2i-1,i =1,2,…,(4n+3) Rule (4)

Case v. n ≡ 4 (mod 6)

f(e2) = 5, f(e3

f(e

) = 3

i

) = 2i-1,i =1, 4, 5,…,(4n+3) Rule (5)

Define f+: V(G) → {0, 1, 2, …, (2k-1)} by f+

Hence the induced map f

(v) ≡ Σ f(uv) mod (2k),

where this sum run over all edges through v …Rule (6)

+ provides the distinct labels for vertices


is

edge - odd graceful.

Example 3.4.2: The connected graph P4 + N7

Proof: The graph P


4 + N7




14 5 11 3 43 1 34

35 37 3941 43 45 47

21 23 25 27 29 31 33

7 9 11 13 15 17 61

49 51 53 55 57 59 19

50 58 4 12 20 28 36


7

Example 3.4.3: The connected graph P4 +N9 is edge - odd graceful.

83

Proof: The graph P4 + N9




60 3 67 1 75 5 2

25 27 29 31 33 35 37 3941 43 45 47 49 51 53 55 57 59

7 9 11 13 15 17 19 21 23

61 63 65 67 69 71 73 75 77

58 66 74 4 12 20 28 36 44


9


Proof: The graph P


4 + N5




36 45 25 43 37 41 8

19 17 15 13 11

9 7 5 3 21

39 37 35 33 31

29 27 25 23 1

Figure 3.17: Edge - odd graceful labeling of P4 42 34 26 18

4 +N

5

84


Proof: The graph P


4 + N6




23 5 44 3 4 1 19

7 9 11 1315 17 19 21 23 25 27 29

31 33 3537 39 41 43 45 47 49 51 53

46 0 8 16 24 32


6


Proof: The graph P


4 + N8


edges, where n ≡ 4 (mod 6). Due to the rules (4) & (6) in theorem (3.4.1)


47 5 38 3 22 1 7

39 41 4345 47 49 51 53

23

25 27 29 31 33 35 37

7 9 11 13 15 17 19 21 55 57 59 61 63 65 67 69

54 62 0 8 16 24 32 40


8

85


Proof: The graph P


4 + N4


edges, where n ≡ 4 (mod6). Due to the rules (5) & (6) in theorem (3.4.1),


5 3 4 5 34 1 23

7 9 11 13

15 17 19 21 23 25 27 29 31 33 35 37

0 8 16 24


4

Section 3.5 - Edge - odd graceful labeling of P5 + Nn:

Theorem 3.5.1: The connected graph P5 + Nn

Proof: Let {v


i: 1 ≤ i ≤ (n + 5)} be the vertices and{ei: 1 ≤ i ≤ (5n+4)}

be the edges of the graph P5+Nn. It is a connected graph such that every

vertex of P5 is adjacent to every vertex of null graph Nn together with

adjacent edges in P5. The arbitrary labelings for vertices and edges

for P5 + Nn


v5 e4 v4 e3 v3 e2 v2 e1 v

1

e5 e6 en+3 en+4 en+5en+6..e2n+3 e2n+4 3n+5 e3n+6…e4n+3e4n+4

e

4n+5 e4n+6….. e5n+3 e

5n+4

e2n+5 e2n+6 ……. e3n+3 e

3n+4

86

v6 v7 v8 ……………………. vn+4 v

n+5


To find edge - odd graceful, define f: E(Pn

5 +Nn ) → {1, 3, …, 2q-1} by

Case i. n ≡ 1(mod 6)

n is odd

f(e1) = 7, f(e2) = 5, f(e3) = 1, f(e4

f(e

) = 3

i

) = 2i-1, i = 5, 6,…,(5n+4) Rule (1)

Case ii. n ≡ 3(mod 6)

f(e1) = 3, f(e2) = 9, f(e5) = 1, f(en+3

f(e

) = 6n+5,

3n+3

f(e

) = 2n+5

i

(3n+2),(3n+4),…,(5n+4) Rule (2)

)=2i-1, i = 3, 4, 6, 7,…,(n+2), (n+4),…,

Case iii. n ≡ 5 (mod 6)

f(e1) = 7, f(e2) = 5, f(e3

f(e

) = 3,

4) = 1, f(en+5

f(e

) = 4n+9,

2n+5) = 2n+9, f(e3n+4) = 10n+7, f(e5n+4

f(e

) = 6n+7

i

(2n+6),…,(3n+3),(3n+5),….,(5n+3) Rule (3)

) = 2i-1, i = 5, 6,…,(n+4), (n+6),…,(2n+4),

Case iv. n ≡ 0 (mod 6)

n is even

f(e1) = 5, f(e2) = 7, f(e3)=3, f(e4

f(e

)=1

i

Case v. n ≡ 2 (mod 6)

) = 2i-1, i =5, 6,…,(5n+4) Rule (4)

f(ei

Case vi. n ≡ 4 (mod 6)

) = 2i-1, i =1, 2,…,(5n+4) Rule (5)

87

f(en+4) = 8n+9, f(e4n+5

f(e

) = 2n+7

i

(4n+6),…,(5n+4) Rule (6)

) = 2i-1, i =1, 2,…,(n+3), (n+5),…,(4n+4),

Define f+: V(G) → {0, 1, 2, …, (2k-1)} by f+

(v) ≡ Σ f(uv) mod (2k),

where this sum run over all edges through v … Rule (7)





Proof: The graph P


5 + N7




30 3 51 1 73 5 21 7 36

2325 2729313335 37 394143454749 51 5355 57596163 65 67 6971 73 75 77

9 11 13 15 17 19 21

29 39 49 59 69 1 11


7

Example 3.5.3: The connected graph P5 + N3 is edge - odd graceful.

88

Proof: The graph P5 +N3




89

6 7 25 5 33 9 23 3 32

1 23 13 15 17 19 21 11 25 27 29 31 33 35 37

21 1 11


3


Proof: The graph P


5 + N5




8 1 13 3 9 5 53 7 20

9 11 13 15 17 29 21 23 25 27 19 31 33 35 57 39 41 43 45 47 49 51 53 55 37

29 39 49 1 11


5


Proof: The graph P


5+ N6




90

17 1 24 3 34 7 40 5 37

21 23 2527 2931 33 3537 3941 43 45 47 495153 55 57 59616365 67

9 11 13 15 17 19

29 39 49

59 1 11


6


Proof: The graph P


5 +N8




47 7 4 5 40 3 76 1 25

41 43454749

31 33 35 37 39 51 5355

9 11 13 15 17 19 21 23 73 75 77 79 81 83 85 87

25 27 29 57 59 61 63 65 67 69 71

29 39 49 59 69 79 1 11


8

91


Proof: The graph P


5+ N4




33 7 44 5 24 3 4 1 7

17 19 21 23 25 27 29 31 33 35 37 39 15 43 45 47

9 11 13 41

Figure 3.27: Edge - odd graceful labeling of P 29 39 1 11

5 +N

4

Section 3.6 - Edge - odd graceful labeling of K1+1Pn:

Definition 3.6.1: K1+Pn is a connected graph obtained from 1 copy of Pn

(whose vertices u1, u2 ,…, un-1, un ; u1, v2, v3, …, vn-1, un is first copy of

Pn) and a null vertex t(n1) whose adjacency edges other than existing

edges are tui (for every i = 1, 2, …,n); uivi

, i = 2, 3, …, (n-1)).

Theorem 3.6.2: The connected graph K1 + 1Pn

Proof: The connected graph K

is edge - odd graceful .

1 + 1Pn has (2n-1) vertices and (4n-4)

edges and the arbitrary labeling for vertices and edges for K1 + 1Pn

are

mentioned below:

92

t

e1 e2 e3 e4…… en-1 e

n

u1 en+1 u2 en+2 u3 en+3 u4 un-2 e2n-2 un-1 e2n-1 un

e

3n-2 e2n e2n+1 e2n+2 e3n-4 e3n-3 e4n-4

v2 e3n-1 v3 e3n v4 vn-2 e4n-5 vn-1

Figure 3.28: Edge - odd graceful labeling of K1+1P

n

To find edge - odd graceful, define f: E(K1+1Pn) → {1, 3, …, 2q-1} by

Case i . n ≡ 0, 2, 4, 6 (mod 8)

n is even

f(ei

) = 2i-1, i = 1, 2, 3,…,(4n-4) Rule (1)

Case (ii). n ≡ 1 (mod 8)

n is odd

f(ei

f(e

) = 2i-1, i = 1, 2, 3,…,(n-1), (n+1),…,(4n-5)

n) = 8n-9, f(e4n-4

) = 2n-1 Rule (2)

Case (iii). n ≡ 3 (mod 8)

f(ei

f(e

) = 2i-1, i = 1, 3, 4,...,(n), (n+2), (n+3),…,(3n-3)

n+1) = 3, f(e2

f(e

) = 2n+1

4n-3-i) = f(e3n-3

)+2i, i=1, 2, 3,….,(n-1) Rule (3)

93

Case (iv). n ≡ 5 (mod 8)

f(ei

f(e

) = 2i-1, i = 1, 2, 3,…,(n-2), (n+1), (n+2),…,(4n-4)

n-1) = 2n-1, f(en

) = 2n-3 Rule (4)

Case (v). n ≡ 7 (mod 8)

f(ei

) = 2i-1, i = 1, 2, 3,…,(4n-4) Rule(5)

Define f+: V(G) → {0, 1, 2, …, (2k -1)} by f+

(v) ≡ Σ f(uv) mod (2k),



and also the edge labeling is distinct. Thus the connected graph K1 + 1Pn


Example 3.6.3: The connected graph K1 + 1P8



1 + 1P8 has 15 vertices and 28 edges,

where n ≡ 0 (mod 8). Due to the rules (1) & (6) in theorem (3.6.2), edge -

odd graceful labeling of the required graph is obtained as follows:

8

1

3 5 7 9 11 13 15

5 17

43 14

19 22 21 30 23 38 25 46 27 54 29 43

31 33 35 37 39 41 55

7 45 13 47 19 49 25 51 31 53 37


8

94

Example 3.6.4: The graph K1 + 1P10



1 + 1P10

has 19 vertices and 36 edges,

where n ≡ 2 (mod 8). Due to the rules (1) & (6) in theorem (3.6.2), edge –


28

1

3 5 7 9 11 13 15 17 19

5 21 14 23 22 25 30 27 38 29 46 31 54 33 62 35 70 37 55

55 39 41 43 71

7 57 45 47 49 51 53

13 59 19 61 25 63 31 65 37 67 43 69 49


10




1 + 1P4

has 7 vertices and 12 edges, where

n ≡ 4 (mod 8). Due to the rules (1) & (6) in theorem (3.6.2), edge - odd

graceful labeling of the required graph is obtained as follows:

16

1

3 5 7

5

9 14 11 22 13 19

19

15 17 23

7

21 13


4

95




1 + 1P6




36

1

3 5 7 9 11

5 5 31

13 14 15 22 17 30 19 38 21

31

23 29 39

25 27

7 33 13 35 19 37 25

Figure 3.32: Edge-odd graceful labeling of K1+1P

6





where n ≡ 1(mod 8). Due to the rules (2) & (6) in theorem (3.6.2), edge -


63

1

3 5 7 9 11 13 15 63

5

19 14 21 22 23 30

49 35 25 38 27 46 29 54 31 62 33 49

37 39 41 43 45 47 17

7

51 13 53 19 55 25 57 31 59 37 61 61

Figure 3.33: Edge - odd graceful labeling of K1+1P9

96




1 + 1P11


where n ≡ 3 (mod 8). Due to the rules (3) & (6) in theorem (3.6.2),


61

1 23 5 7 9 11 13 15 17 19 21

3 3

14 25 22 27 30 29 38 31 46 33 54 35 62 37 70 39 78 41 43

79 43 45 47 49 51 53 55 57 59 61

39 77 37 75 35 73 33 71 31 69 29 67 27 65 25 63 23


11




1 + 1P5


n ≡ 5 (mod 8). Due to the rules (4) & (6) in theorem (3.6.2), edge - odd


25

1

3 5 9 7

5

11

14 13 22 15 0 17 23

25

19 21 23 31

7 27 13 29 19


5

97




1 + 1P7


where n ≡ 7(mod 8). Due to the rules (5) & (6) in theorem (3.6.2),


1

1

3 5 7 9 11 13

5

15 14 17 22 19 30 21 38 23 46 25 37

37 27 29 31

33 35 47

7 39 13 41 19 43 25 45 31


7

Section 3.7 - Edge - odd graceful labeling of K1+2Pn

Definition 3.7.1: K:

1 + 2Pn is a connected graph obtained from 2 copies

of Pn (whose vertices u1, u2, …, un-1, un; u1, v2, v3, …, vn-1, un is first

copy of Pn; u1, w2,….,wn-1, un is second copy of Pn) and a null vertex

t(n1) whose adjacency edges other than existing edges are tui (for every

i =1,2, …., n); uivi, i = 2, 3, …, (n-1); vjwj

, j = 2, 3, …, (n-1)).

Lemma 3.7.2: The graph K1 + 2Pn

Proof: The graph K

is edge - odd graceful where n = 2, 4.

1 + 2P2


edges. The arbitrary labeling of edge- odd graceful of the required graph


98

6

5 1

3

4 7 0

9


2

The graph K1 + 2P4




16

7

3 5 1

30

9 4 11 12 13 2

19

15

17 23

29 14 21 18 33

27 25

19 31 21


4





edges and the arbitrary labelings for vertices and edges for K1 + 2Pn are

mentioned below:

99

t

e1 e2 e3 …. en-2 en-1 e

n

u

un

1 en+1 e

e2n-1

3n-2 u2 e2n-2 un-1 e4n-4 e

e6n-7

5n-5 e2n en+2 u3 un-2 e3n-3

v

2 en+3 u4 u5 e3n-4 vn-1

w

2 e4n-3 e2n+1 e2n+2 vn-2 e4n-5 e5n-6 wn-1

e

3n-1

v

3 e3n v4 v5 e5n-7 e

e6n-8

5n-4 e4n-2 e5n-8 e6n-9 wn-2

w

3 e4n-1

e

5n-3 w4 e5n-2 w

5


n


Case i . n ≡ 0 (mod 6)

n is even

f(ei

(5n-4),…,(6n-7)

) = 2i-1, i = 2, 3,…,(n-1), (n+2),…, (4n-4), (5n-5),

f(en) =1; f(e1) = 2n+1; f(en+1

f(e

) = 2n-1

5n-5-i) = f(e4n-4

)+2i, i = 1, 2, 3, ….,(n-2) Rule(1)

Case ii . n ≡ 2 (mod 6)

f(ei

(3n-1),(4n-4),(5n-5),(5n-4),…,(6n-7)

) = 2i-1, i = 2, 3,…,(n-1), (n+1), (n+2),…,(3n-3),

f(en) =1; f(e1) = 6n-5 ; f(e3n-2

f(e

) =2n-1

5n-5-i)= f(e4n-4)+2i, i = 1, 2, 3,….,(n-2) Rule(2)

100

Case iii . n ≡ 4 (mod 6)

f(ei

(5n-4),…, (6n-7) Rule(3)

) = 2i-1, i = 2,3,…,(n-1),(n+1) (n+2),…, (4n-4),

f(en) =1; f(e1) =10n-11 ; f(e5n-5

f(e

) =2n-1

5n-5-i) = f(e4n-4

)+2i, i=1, 2, 3,….,(n-2)

Case iv. n ≡ 1, 3, 5 (mod 6)

n is odd

f(ei

Define f

) = 2i-1, i = 1, 2, 3, …, (6n-7) Rule(4)

+: V(G) → {0, 1, 2, …, (2k -1)} by f+

(v) ≡ Σ f(uv) mod (2k),



and also the edge labeling is distinct. So the connected graph K1 + 2Pn

is

edge - odd graceful.




1 + 2P6




38

13

3 5 7 9 1

46 21 2

11 52 15 4 17 12 19 20

31 23

25 27 29 39

18

30

49 33 22 35 26 37 41 57

47 45

43 37

31 51 33 53 35 55

Figure 3.40: Edge - odd graceful labeling of K

1+2P6

101




1 + 2P8




10

43 3 5 7 9 11 13 1

62 17 70 19 78 21 4 23 12 25 20 27 28 29 2

15 31 33 35 37 39 41 55 81

69 76 45 26 47 30 49 34 51 38 53 42

67 65 63 61 59 57

43 71 45 73 47 75 49 77 51 79 53


8




1 + 2P10




102

64

89 3 5 7 9 11 13 15 17 1

78 21 86 23 94 25 102 27 4 29 12 31 20 33 28 35 36 37 2

19 55 39 41 43 45 47 49 51 53 71

26 57 30 59 34 61 38 63 42 65 46 67 50 69 54

87 85 83 81 79 77 75 73 105

91 91 57 93 59 95 61 97 63 99 65 101 67 103 69

Figure 3.42: Edge- odd graceful labeling of K1+2P

10



Proof: The connected graph K1 + 2P7 has 18 vertices and 35 edges,

where n ≡ 1(mod 6). Due to the rules (4) & (5) in theorem (3.7.3), edge -


49

1

3 5 7 9 11 13

14

42 15 62 17 0 19 8 21 16 23 24 25

37 27

29 31 33 35 47 69

59 45 44

12 39 20 41 28 43

36

49 51

53 55 57

65 47

29 61 35 63 41 65 47 67 53


103







81

1 3 5 7 9 11 13 15 17

54 19 78 21 86 23 0 25 8 27 16 29 24 31 32 33 18

49 35 37 39 41 43 45 47 63 93

79 12 51 20 53 28 55 36 57 44 59 52 61 60

65 67 69 71 73 75 77

37 81 43 83 49 85 55 87 61 89 67 91 73


9







25

1

3 5 7 9

30

11 0 13 8 15 16 17 10

25

19

21 23 31

39 12 27 20 29 28 45

33 35

37

21 41 27 43 33


1+2P

5

104

Section 3.8 - Edge - odd graceful labeling of K1 + 3Pn

Definition 3.8.1: K

:


of Pn (whose vertices u1, u2, …, un-1, un; u1, v2, v3, …, vn-1, un is first

copy of Pn ; u1, w2,….,wn-1, un is second copy of Pn; u1, s2,….,sn-1, un is

third copy of Pn ) and a null vertex t (n1) whose adjacency edges other

than existing edges are tui (for every i = 1, 2,…,n); uivi, i= 2,3,….,(n-1);

vjwj, j = 2, 3 ,…, (n-1); wksk

, k = 2, 3,…,(n-1)).





edges and the arbitrary labelings for vertices and edges for K1+3Pn

are

mentioned below:

t

en en-1 en-2 …. e3 e2 e

1

un e2n-1 un-1 e2n-2 un-2 e2n-3 u3 en+2 u2 en+1 u

e1

3n-3 e3n-4 e3n-5 e2n+1 e2n e3n-2

e

4n-4 vn-1 e4n-5 vn-2 e4n-6 vn-3 v3 e3n-1 v

e2

6n-7 e5n-6 e5n-7 e5n-8 e4n-2 e4n-3 e

5n-5

e8n-10 wn-1 e6n-8 wn-2 e6n-9 wn-3 w3 e5n-4 w2 e

e7n-8

7n-9 e7n-10 e7n-11 e6n-5 e6n-6

s

n-1e8n-11 sn-2 e8n-12 sn-3 s3 e7n-7 s

2


n

105


Case i . n ≡ 2, 4, 6, 8 (mod14)

n is even

f(ei

f(e

)=2i-1,i= 2, 3,…,(n-1), (n+1),(n+2),…, (8n-10) Rule(1)

n) =1; f(e1

Case ii . n ≡ 10,12,14 (mod14)

) =2n-1

f(ei

(5n-4),…,(8n-10)

) = 2i-1, i = 2, 3,…, (n+1), (n+2),…,(5n-6),

f(e1) = 10n-11; f(en) =1; f(e5n-5

) =2n-1 Rule(2)

f(e

n is odd

i

(n+2), …, (8n-10) Rule(3)

) = 2i-1, i = 2, 3,…, (n-1), (n+1),

f(e1) = 2n-1; f(en

) =1

Define f+: V(G) → {0, 1, 2, …, (2k -1)} by f+

Hence the induced map f

(v) ≡ Σ f(uv) mod (2k),


+ provides the distinct labels for vertices

and also the edge labeling is distinct. Thus the connected graph K1 + 3Pn


Example 3.8.3: The connected graph K1 + 3Pn


is edge-odd graceful,

where n ≡ 2, 4, 6, 8 (mod14).

1 + 3P2 has 3 vertices and 6 edges. Due to

the rules (1) & (4) in theorem (3.8.2), edge - odd graceful labeling of the

required graph is obtained as follows:

11

3

4 5 7 9 11

1

Figure 3.47: Edge - odd graceful labeling of K 9

1+3P2

106

The connected graph K1 + 3P4 has 11 vertices and 22 edges. Due

to the rules (1) & (4) in theorem (3.8.2), edge - odd graceful labeling of

the required graph is obtained as follows:

16

1

5 3 7

25 13

2 11 38 9 15

23

17

0 21

15 19

43 33 27 36 25 29 39

40 31 32

37 35

33 41 27


The connected graph K

4

1 + 3P6 has 19 vertices and 38 edges. Due



36

1

9 7 5 3 11

41

21 2 19 70 17 62 15 54 13 19

39

29 27 25 23 31

75 57 0 37 68 35 60 33 52 49 67

47 45

43 41

72 55 64 53 56 51 48

65 63 61 59

61 73 55 71 49 69 43


107




64

1 13 11 9 7 5 3 15

57 29 2 27 102 25 94 23 86 21 78 19 70 17 23

55 41

39 37 35 33 31 43

81

0 53 100 51 92 49 84 47 76 45 68 69 95

67 65

104 63 61 59 57

107 79 96 77 88 75 80 73 72 71 64

93 91 89 87 85 83

89 105 83 103 77 101 71 99 65 97 59


8

Example 3.8.4: The connected graph K1 + 3Pn


is edge - odd graceful where n ≡ 10, 12, 14 (mod14).

1 + 3P10 has 35 vertices and 70 edges. Due to the rules (2) & (4) in theorem (3.8.2), edge - odd graceful labeling of the required graph is obtained as follows:

30

1

17 15 13 11 9 7 5 3 89

73 37 2 35 134 33 126 31 118 29 110 27 102 25 94 23 86 21 27

71 53 51 49 47 45 43 41 39 55

105

139 0 69 132 67 124 65 116 63 108 61 100 59 92 57 84 19 123

87 85 83 81 79 77 75 73

136 103 128 101 120 99 112 97 104 95 96 93 88 91 10

121 119 117 115 113 111 109 107

117 137 111 135 105 133 99 131 93 129 87 127 81 125 75


108

The connected graph K1 + 3P12 has 43 vertices and 86 edges. Due to the rules (2) & (4) in theorem (3.8.2), edge - odd graceful labeling of the required graph is obtained as follows:

52

1 21 19 17 15 13 11 9 7 5 3 109

89 45 2 43 166 41 158 39 150 37 142 35 134 33 126 31 118 29 110 27 102 25 31

87 65 63 61 59 57 55 53 51 49 47 67

0 85 164 83 156 81 148 79 140 77 132 75 124 73 116 71 108 69 100 23

129 107 105 103 101 99 97 95 93 91 89

168 127160 125 152 123 144 121 136 119 128 117 120 115 112 113 104 111 10

171 149 147 145 143 141 139 137 135 133 131 151

145 169 139 167 133 165 127 163 121 161 115 159 109 157 103 155 97 153 91


12




94

1 25 23 21 19 17 15 13 11 9 7 5 3 129

35

105 53 2 51 198 49 190 47 182 45 174 43 166 41 158 39 150 37142 35 134 33 126 31 118 29

103 77 75 73 71 69 67 65 63 61 59 57 55 79

0 101 196 99 188 97 180 95 172 93 164 91 156 89 148 87 140 85 132 83 124 81 116 27

153 127 125 123 121 119 117 115 113 111 109 107 105 179

200 151 192 149 184 147 176 145 168 143 160 141 152 139 144 137 136 135 128 133 120 131 10

203

177 175 173 171 169 167 165 163 161 159 157 155

173 201 167 199 161 197 155 195 149 193 143 191 137 189 131 187 125 185 119 183 113 181 107


109


Definition 3.9.1: K:


of Pn (whose vertices u1, u2,…,un-1, un ; u1, v2, v3,…,vn-1, un is first copy

of Pn; u1, w2,….,wn-1, un is second copy of Pn; u1, s2,….,sn-1, un is third

copy of Pn; u1, x2,….,xn-1, un is fourth copy of Pn) and a null vertex t (n1)

whose adjacency edges other than existing edges are tui (for every i = 1,

2,…,n); uivi, i=2, 3,….,(n-1); vjwj, j = 2, 3,…,(n-1), wksk, k = 2,

3,…,(n-1); slxl

, l = 2, 3,…,(n-1)).

Lemma 3.9.2: The connected graph K1 + 4P2

Proof: The graph K


1 + 4P2




4

5

8 1 7 9 11 13 3

2


2



is edge – odd graceful .


edges and the arbitrary labelings for vertices and edges for K1 + 4Pn

are

mentioned below:

110

n is even

t

en en-1 en-2 …e3 e2 e1

un e2n-1 un-1 e2n-2 e2n-3 4 u3 en+2 u2 en+1 u

e1

4n e3n-3 e3n-4 e3n-5 e2n+1 e2n e3n-2

v

n-1 e4n-1 vn-2 e4n-2 v3 e3n-1 v

e2

6n-7 e5n-6 e5n-7 e5n-8 e4n+2 e4n+1 e

e5n-5

8n-10 wn-1 e6n-8 wn-2 e6n-9 w3 e5n-4 w2 e

e7n-8

7n-9 e7n-10 e7n-11 e6n-5 e6n-6

s

n-1 e8n-11 sn-2 e8n-12 s3 e7n-7 s

e2

10n-13 e9n-12 e9n-13 e9n-14 e8n-8 e8n-9 e

9n-11

xn-1 e10n-14 xn-2 e10n-15 x3 e9n-10 x

Figure 3.55: Edge - odd graceful labeling of K2

1+4P

n

n is odd

e10n-3 e10n-4 …. e9n-1e9n-2e1

un en+2 un-1 en+3 un-2 en+4 u3 e2n-2 u2 e2n-1 u

e1

2n e2n+1 e2n+2 e3n-4 e3n-3 e4n-4

e

v3n-2

n-1 e3n-1 vn-2 e3n vn-3 v3 e4n-5 v2 e6n-1

e

5n e4n-3 e4n-2 e4n-1 e5n-1

w

n-1 e5n+1 wn-2 e5n+2 wn-3 w3 e6n-2 w2e8n-2 e

7n-1 e6n e6n+1 e6n+2 e7n-3 e7n-2

s

n-1 e7n sn-2 e7n+1 sn-3 s3 e8n-3 s

e2

2 e8n-1 e8n e8n+1 e9n-4 e9n-3 e

n+1

xn-1 e3 xn-2 e4 xn-3 x3 en x


2

1+4Pn

111

To find edge - odd graceful, define f: E(K1+4Pn

n is even

) → {1, 3, …, 2q-1} by

f(ei

f(e

) = 2i-1, i = 2, 3,…,(n-1), (n+1),(n+2),…, (10n-13) Rule(1)

n) =1; f(e1

) =2n-1

n is odd

Case i: n ≡ 1 (mod 4)

f(ei

) = 2i-1, i = 1, 2,…,(10n-13) Rule(2)

Case ii: n ≡ 3 (mod 4)

f(ei

) = 2i-1, i = 2, 3,…,(10n-14) Rule(3)

f(e1) = 20n-27; f(e10n-13

) =1

Define f+: V(G) → {0, 1, 2, …, (2k -1)} by f+

(v) ≡ Σ f(uv) mod (2k),

where this sum run over all edges through v …. Rule (4)


and also the edge labeling is distinct. Therefore the connected graph

K1+4Pn





1+4P6


n is even. Due to the rules (1) & (4) in theorem (3.9.3), edge - odd


112

36

1 9 7 5 3 11

4 21 78 19 70 17 62 15 54 13 68

39

29 27 25 23 31

57 58 37 50 35 42 33 34

49

47 45 43 41

75 36 55 28 53 20 51 12 67

65 63 61 59 85

93 14 73 6 71 92 69 84

83 81 79 77

Figure 3.57: Edge-odd graceful labeling of K 79 91 73 89 67 87 61

1+4P

6



is edge - odd graceful where

n ≡ 1 (mod 4).

1 + 4P5 has 18 vertices and 37 edges. Due



59

73 71 69 67 1

56 11 40 13 44 15 48 17 14

25

19 21 23 31

39 30 27 38 29 46 45

33 35 37

53 12 41 20 43 28 59

47 49 51

3 68 55 2 57 10

61 63 65 9

69 5 1 7 7


1+4P5

113


is edge-odd graceful where n ≡ 3

(mod 4).

Proof: The connected graph K1 + 4P7




79

1 111 109 107 105 103 113

82 15 56 17 60 19 64 21 68 23 72 25 14

37

27 29 31 33 35 47

59 38 39 46 41 54 43 62 45 70 69

49 51 53 55 57 91

81 12 61 20 63 28 65 36 67 44

71 73 75 77 79 13

3 100 83 108 85 2 87 10 89 18

93 95 97 99 101

101 5 107 7 113 9 5 11 11

Figure 3.59: Edge - odd graceful labeling of K1 + 4P

7


Definition 3.10.1: K

:


of Pn (whose vertices u1, u2,…,un-1, un; u1, v2, v3,…,vn-1, un is first copy

of Pn; u1, w2,….,wn-1, un is second copy of Pn; u1, s2,….,sn-1, un is third

copy of Pn; u1, x2, ..., xn-1, un is fourth copy of Pn; u1, y2, …, yn-1, un is

fifth copy

114

of Pn) and a null vertex t (n1) whose adjacency edges other than existing

edges are tui (for every i = 1, 2,…,n); uivi, i = 2, 3,….,(n-1), vjwj,

j = 2, 3,…,(n-1); wksk, k = 2, 3,…,(n-1); slxl, l = 2, 3,…,(n-1); xmym

,

m = 2, 3,…,(n-1)).

Lemma 3.10.2: The connected graph K1 + 5P4



1 + 5P4

has 15 vertices and 32 edges. The

arbitrary edge-odd graceful labeling of the required graph is obtained as

follows:

10

7

5 3 59

41 13 46 11 38 9 13

23 17

15 19

33 24

21 16 29

43 27 25

39 49 1

51 0 31 56

37 41 35

40 32

47 45

14 53 12

63 55 57

Figure 3.60: Edge - odd graceful labeling of K51 61 55

1+5P

4

Theorem 3.10.3: The graph K1 + 5Pn




edges and the arbitrary labelings for vertices and edges for K1+5Pn

are

mentioned below:

115

n is even en en-1 en-2 ……. e3 e2 e1

un e2n-1 un-1 e2n-2 e2n-3 u3 en+2 u2 en+1 u

e1

4n-4 e3n-3 e3n-4 e3n-5 e2n+1 e2n e3n-2

e

6n-7 vn-1 e4n-5 vn-2 v3 e3n-1 v

2

e5n-6 e5n-7 e4n-2 e4n-3 e

e5n-5

8n-10 wn-1 e6n-8 wn-2 e6n-9 w3 e5n-4 w2 e7n-8

e

10n-13 e7n-9 e7n-10 e7n-11 e6n-5 e

e6n-6

12n-16 sn-1 e8n-11 sn-2 e8n-12 s3 e7n-7 s

e2

9n-12 e9n-13 e9n-14 e8n-8 e8n-9 e

x9n-11

n-1 e10n-14 xn-2 e10n-15 x3 e9n-10 x2 e

e11n-14

11n-15 e11n-16 e10n-11 e

y10n-12

n-1 e12n-17 yn-2 y3 e11n-13 y

Figure 3.61: Edge - odd graceful labeling of K2

1+5P

n

n is odd

en en-1 en-2 ………. e3 e2 e

1

un e3n-2 un-1 e3n-3 e3n-4 u3 e2n+1 u2 e2n u

e1

4n-3 e8n-8 e8n-9 e7n-4 e7n-5 e3n-1

v

n-1 e4n-4 vn-2 v3 e3n v

e2

5n-4 e9n-10 e9n-11 e8n-6 e8n-7 e

e4n-2

6n-5 wn-1 e5n-5 wn-2 w3 e4n-1 w2 e

e5n-3

10n-12 e10n-13 e9n-8 e9n-9 e

e6n-4

7n-6 sn-1 e6n-6 sn-2 s3 e5n-2 s

e2

11n-14 e11n-15 e10n-10 e

e10n-11

x2n-1

n-1 e7n-7 xn-2 x3 e6n-3 x2

e

12n-16 e12n-17 e11n-12 e11n-13 e

yn+1

n-1 e2n-2 yn-2 y3 en+2 y2

Figure 3.62: Edge - odd graceful labeling of K1+5Pn

116

To find edge - odd graceful, define f: E(K1+5Pn

n is even

) → {1, 3, …, 2q-1} by

Case i . n ≡ 0 (mod 4), n≠ 4

f(ei

f(e

) = 2i-1,i = 2,3,…, (11n-15), (11n-13)…, (12n-16) Rule(1)

11n-14) = 1; f(e1

) = 22n-29

Case ii . n ≡ 2 (mod 4)

f(ei

) = 2i-1, i = 1, 2, 3,…,(12n-16) Rule(2)

n is odd

f(ei

Define f

) = 2i-1, i = 1, 2, 3,…,(12n-16) Rule(3)

+: V(G) → {0, 1, 2, …, (2k -1)} by f+

(v) ≡ Σ f(uv) mod (2k),



and also the edge labeling is distinct. Therefore the connected graph

K1 + 5Pn




Proof: The connected graph K1 + 5P8




117

50

15 13 11 9 7 5 3 147

99 29

110 27 102 25 94 23 86 21 78 19 70 17 13

55 41 39

37 35 33 31 43

56

53 48 51 40 49 32 47 24 45 16 69

159 81 67 65 63 61 59 57 95

133107 0

79 152 77 144 75 136 73 128 71 120 121

93 91 89 87 85 83 95 1

104 105 96 103 88 101 80 99 72 97 64

119 117 115 113 111 109

38 131 34 129 30 127 26 125 22 123 18

135 137 139 141 143 145

131 157 129 155 127 153 125 151 123 149 135


Example 3.10.5: The K8

1 + 5P6

Proof: The connected graph K is edge-odd graceful where n ≡ 2 (mod 4). 1 + 5P6

has 27 vertices and 56 edges. Due to the rules (2) & (4) in theorem (3.10.3), edge - odd graceful labeling of the required graph is obtained as follows:

36

11

9 7 5 3 1

71

21 78 19 70 17 62 15 54 13 13

39

29 27 25 23 31 49

75 57 40 37

32 35 24 33 16 67

111 93 47 45 43 41

0 55 104 53 96 51 88 85

65 63 61 59 103

72 73 64 71 56 69 48

83 81 79 77

26 91 22 89 18 87 14

95 97 99 101

Figure 3.64: Edge - odd graceful labeling of K 91 109 89 107 87 105 85

1+5P

6

118



is edge-odd graceful, if n is odd.

1 + 5P5

has 21 vertices and 44 edges. Due



25

9 7 5 3 1

21 25

30 23 22 21 14 19 11

4941 33

63 61 59 27

108 31 67 100 29 92 35

69 67 65 43

1757 48 39 40 37 32

75 73 71

76 47 68 45 60 51

81 79 77

104 55 96 53 88 11

87 85 83

Figure 3.65: Edge - odd graceful labeling of K31 15 25 13 19

1+5P

5

s

CHAPTER III SUM OF PATH AND NULL GRAPH -...

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