Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 47, No. 1 (2025) pp. 133-152

Valani Darshana K¹

EDGE ODD GRACEFUL LABELING OF

SOME SNAKE GRAPHS

and

Kanani Kailas K²

Abstract: An edge odd graceful labeling of graph

G

is a bijection

f

from

the edges of the graph to {1, 3, ..., 2q  1} such that, when each vertex is

assigned the sum of all the edges incident to it mod 2q the resulting vertex

labels are distinct. A graph is called an edge odd graceful graph as it admits

an edge odd graceful labeling. It was intoduced by Solairaju and Chithra in

2008. In this research paper, Edge odd graceful labeling of some snake

graphs such as double alternate triangular snakeDA(T_n), double alternate

quadrilateral snake DA(Q_n) and alternate pentagonal snake A(PS_n) have

been discussed.

Keywords: Edge Odd Graceful Labeling, Edge Odd Graceful Graph, Snake

Graphs.

Mathematics Subject Classification (2000) No.: 05C78.

1. Introduction

In this research article, all graphs G  (V(G), E(G)) are finite, simple,

connected and undirected. Here V(G) be the vertex set and E(G)be the edge set of a

graph. Graph labeling is an assignment of integers to edges or vertices or both,

subject to certain conditions. For an extensive survey on graph labeling and

134

VALANI DARSHANA K AND KANANI KAILAS K

bibliographic references, we refer to Gallian [2]. A graceful labeling of a graph

G

,

which was introduced by Rosa [6] in 1967, is a injection from the vertices of the

f

graph to the set {1, 2, ..., q} such that the induced function f^from the set of edges

to the set {0, 1, 2, ..., q} defined as f^(e  uv)  f(u)  f(v)  , is bijective.

Soleha et al. [10] have proved that the alternate triangular snake and alternate

quadrilateral snake graphs are edge odd graceful.

Definition 1.1 [9]: A function

f

is called an edge odd graceful labeling

is bijective and the induced

of a graph if f : E(G)  {1, 3, ..., 2q  1}

G

function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined as f^(u) 

f(uv)

_uvE(G)

(mod 2q) is injective.

A graph which admits an edge odd graceful labeling is called an edge

odd graceful graph.

Definition 1.2 [1]: An alternate triangular snake

A(T_n) is obtained

from a path with vertices u₁, u₂, ..., u_nby joining u_iand u_i 1

P

n

(alternatively) to new vertex v_i, where 1  i  n  1 for even n and for

1  i  n  2 for odd n.

That is every alternate edge of a path

P_nis replaced by C₃

.

Definition 1.3 [1]: A double alternate triangular snake DA(T_n) is

obtained from a path with vertices u₁, u₂, ..., u_nby joining u_iand u_i1

P

n

(alternatively) to two new vertices v_iand w_i, where 1  i  n  1 for even

n and for 1  i  n  2 for odd n.

In other words, the double alternate triangular snake DA(T_n) consists

of two alternate triangular snakes that have a common path.

Definition 1.4 [1]: An alternate quadrilateral snake A(Q_n) is obtained

from a path

P

with vertices u₁, u₂, ..., u_nby joining u_iand u_i1

n

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

135

(alternatively) to two new vertices v_iand w_i, respectively and then joining v_i

,

and w_iwhere 1  i  n  1 for even n and for 1  i  n  2 for odd n.

That is every alternate edge of a path

P

is replaced by C₄

.

n

Definition 1.5 [1]: The double alternate quadrilateral snake DA(Q_n)

obtained from a path with vertices u₁, u₂, ..., u_nby joining u_iand u_i 1

(alternatively) to two new vertices v_i

joining v_iv_i1and w_iw_i1, where 1  i  n  1 for even n and for

P

n

,

w_iand v_i1

,

w_i1respectively and then

,

1  i  n  2 for odd n .

In other words, the double alternate quadrilateral snake graph

DA(Q_n) consists of two alternate quadrilateral snakes that have a common

path.

Definition 1.6 [8]: An alternate pentagonal snake A(PS_n) is obtained

from a path

P

with vertices u₁, u₂, ..., u_nby joining u_iand u_i 1

n

(alternatively) to new vertices v_iand w_irespectively and then joining v_iand

w_ito the new vertex x_i, where 1  i  n  1 for even n and for

1  i  n  2 for odd n.

That is, every alternate edge of path

P_nis replaced by a cycle C₅

.

2. Main Results

Theorem 2.1: The double alternate triangular snake DA(T_n) is an edge

odd graceful graph for all n  2

.

Proof: Let

G

be a double alternate triangular snake DA(T_n) which is

_nwith vertices u₁, u₂, ..., u_nby joining u_iand u_i 1

obtained from a path

P

(alternatively) to two new vertices v_jand w_j, where 1  i  n  1 for even

n

,

n

₂

1  i  n  2 for odd

n

and 1  j 

.

136

VALANI DARSHANA K AND KANANI KAILAS K

Therefore

n

₂

V(G)  {u_i, v_j, w_j/ 1  i  n, 1  j 

}

n

₂

n

₂

E(G)  {u_iu_i1/1  i n  1}  {u_2i1v_i/1  i 

}  {u v /1  i 

}

2i i

n

₂

n

₂

 {u_2i1w_i/1  i 

}  {u w /1  i 

}

.

2i

i

Here note that

2n,

if n  0 (mod 2)



V(G) 



2n  1, if n  1 (mod 2)





3n  1, if n  0 (mod 2)



E(G) 



3n  3, if n  1 (mod 2)





Case 1: n  0, 2 (mod 4)

Subcase 1: n  2

v₁

4

3

7

1

5

9

5

9

u₁

u₂

2

w₁

Figure 1: Edge odd graceful labeling of double alternate triangular snake DA(T₂)

Here Figure 1 shows that the double alternate triangular snake DA(T₂) is an

edge odd graceful graph.

Subcase 2:n  0, 2 (mod 4) and n  2

Define edge labeling f : E(G)  {1, 3, 5, ..., 6n  3} is as follows:

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

137

f(u_iu_{i 1})  4n  2i  1; 1  i  n  1

f(u_{2i 1}v_i)  4i  3; 1  i  ⁿ

2

f(u_2iv_i)  4i  1; 1  i  ⁿ

2

f(u_2i1w_i)  6n  4i  1; 1  i  ⁿ

2

f(u_2iw_i)  6n  4i  1; 1  i  ⁿ

2

The corresponding labels of vertices u_iand v_i, i  1, 2, 3... mod (6n  2)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  f(u₁w₁)  4n  3

;

f^(u_i)  f(u_i1u_i)  f(u_iu_i1)  f(u_iv_₂_i_)  f(u_iw_₂_i_

)

 2n  4i  2; 2  i  n  1

f^(u_n)  f(u_{n 1}u_n)  f(u_nv )  f(u_nw )  2n  1

;

n

2

n

2

n

2

f^(v_i)  f(u_2i1v_i)  f(u_2iv_i)  8i  4; 1  i 

n

2

f^(w_i)  f(u_2i1w_i)  f(u_2iw_i)  6n  8i  2; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 6n  3} . Then the labels of

vertices are in the set

{4, 12, ..., 4n  4}  {4n  3}  {2n  6, 2n  10, ..., 4n  4}  2n  1

 6n  6, 6n  14, ..., 2n  2

.

Here

i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1} , defined

as f^(u) 

f(uv) mod (6n  2) is injective.

_uvE(G)

138

VALANI DARSHANA K AND KANANI KAILAS K

Case 2: n  1 (mod 4)

Define edge labeling f : E(G)  {1, 3, 5, ..., 6n  7} is as follows:

f(u_iu_{i 1})  4n  2i  3; 1  i  n  2

f(u_{n 1}u_n)  2n  1

;

ⁿ₂

f(u_2i1v_i)  4i  3; 1  i 

ⁿ₂

f(u_2iv_i)  4i  1; 1  i 

ⁿ₂

f(u_2i1w_i)  4n  4i  1; 1  i 

ⁿ₂

f(u_2iw_i)  4n  4i  1; 1  i 

The corresponding labels of vertices u_iand v_i, i  1, 2, 3... mod (6n  6)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  f(u₁w₁)  8n  3

;

f^(u_i)  f(u_i1u_i)  f(u_iu_i1)  f(u_iv

i

_₂_)  f(u_iw _₂_

i

)

 6n  4i  4; 2  i  n – 2

f^(u_n1)  f(u_{n2 n1}

u

)  f(u_n1u_n)  f(u_n1v_{ })  f(u_nw_{ })  6n  4

n

2

n

2

f^(u_n)  f(u_n1u_n)  2n  1

;

ⁿ₂

f^(v_i)  f(u_2i1v_i)  f(u_2iv_i)  8i  4; 1  i 

ⁿ₂

f^(w_i)  f(u_{2i 1}wi)  f(u_2iw_i)  2n  8i  6; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 6n  7}. Then the labels of

vertices are in the set {4, 12, ..., 4n  8}  2n  3  6n  4, 6n  8, ..., 4n  6

 2n  1  2n  2, 2n  10, ..., 4n  4

.

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

Here i  j, f(v_i)  f(v_j)

139

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

_uvE(G)f(uv) mod (6n  6) is injective.



Case 3: n  3 (mod 4)

Define edge labeling f : E(G)  {1, 3, 5, ..., 6n  7} is as follows:

f(u_iu_i1)  2n  2i  3; 1  i  n – 1

ⁿ

2



f(u_{2i 1}v_i)  4i  3; ; 1  i 

ⁿ

2



f(u_2iv_i)  4i  1; 1  i 

ⁿ

2



f(u_{2i 1}w_i)  4n  4i  7; 1  i 

ⁿ

2



f(u_2iw_i)  4n  4i  5; 1  i 

The corresponding labels of vertices u_iand v_i, i  1, 2, 3... mod (6n  6)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  f(u₁w₁)  6n  3

;

f^(u_i)  f(u_i1u_i)  f(u_iu_i1)  f(u_iv

i

_₂_)  f(u_iw _₂_

i

)

 2n  8i  8; 2  i  n – 1

f^(u_n)  f(u_n1u_n)  4n  5

;

ⁿ

2



f^(v_i)  f(u_2i1v_i)  f(u_2iv_i)  8i  4; 1  i 

ⁿ

2



f^(w_i)  f(u_2i1w_i)  f(u_2iw_i)  ; 1  i 

140

VALANI DARSHANA K AND KANANI KAILAS K

The labels of edges are in the set {1, 3, 5, ..., 6n  7}. Then the labels of

vertices

are

in

the

set

{4, 12, ..., 4n  4}  6n  3

 2n  8, 2n  16, ..., 4n  10  4n  5  {2n  2, 2n  10, ..., 6n  10}

.

Here i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

f(uv) mod (6n  6) is injective.

_uvE(G)

Example 2.2: The edge odd graceful labeling of double alternate

triangular snake DA(T₈) DA(T₉) and DA(T₁₁) is shown in Figure 2, 3, and 4.

,

Figure 2: The edge odd graceful labeling of double alternate triangular

snake DA(T₈)

Figure 3: The edge odd graceful labeling of double alternate triangular

snake DA(T₉)

Figure 4: The edge odd graceful labeling of double alternate triangular

snake DA(T₁₁)

.

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

141

In each possibility the graph under consideration satisfies the vertex

conditions and edge conditions for an edge odd graceful labeling. Hence, the double

alternate triangular snake DA(T_n) is an edge odd graceful graph for all n  2

.



Theorem 2.3: The double alternate quadrilateral snake DA(Q_n) is an

edge odd graceful graph for all n  2

.

Proof: Let

obtained from a path

(alternatively) to two new vertices v_i

joining v_i

v_i1and w_iw_i1, where 1  i  n  1 for even

G

be a double alternate quadrilateral snake DA(Q_n) which is

_nwith vertices u₁, u₂, ..., u_nby joining u_iand u_{i 1}

w_iand v_i1w_i1respectively and then

and for

P

,

n

1  i  n  2 for odd

n

.

Therefore,

V(G)  {u_i, v_i, w_i/1  i  n}

ⁿ

2



E(G)  {u_iu_i1/1  i  n  1}  {v_2i1v_2i/1  i 

}

ⁿ

2



 {w_2i1w_2i/1  i 

}  {u v /1  i  n}  {v w /1  i  n}

.

i i

i

Here note that

3n,

if n  0 (mod 2)



V(G) 



3n  2, if n  1 (mod 2)





4n  1, if n  0 (mod 2)



E(G) 



4n  4, if n  1 (mod 2)





Case 1: n  0 ( mod 2)

Subcase 1: n  2

142

VALANI DARSHANA K AND KANANI KAILAS K

Figure 5: Edge odd graceful labeling of double alternate quadrilateral snake

graph DA(Q₂)

Here Figure 5 shows that double alternate quadrilateral snake DA(Q₂) is an

edge odd graceful graph.

Subcase 2: n  0 ( mod 2) and n  2

Define edge labeling f : E(G)  {1, 3, 5, ...8n  3} is as follows:

f(u_iu_{i 1})  3n  2i  1; 1  i  n – 1

n

2

f(u_{2i 1 2i1})  6i  5; 1  i 

v

f(u_2iv_2i)  6i  1; 1  i  ⁿ

2

f(v_iv_i1)  6i  3; 1  i  ⁿ

2

n

2

f(u_2i1

w

_2i1)  8n  6i  3; 1  i 

n

2

f(u_2iw_2i)  8n  6i  1; 2  i 

n

2

f(w_iw_i1)  8n  6i  1; 1  i 

The corresponding labels of vertices u_iand v_i

,

i  1, 2, 3... mod (8n  2)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  f(u₁w₁)  3n  1;

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

143

f^(u_i)  f(u_i1u_i)  f(u_iu_i1)  f(u_iv_i)  f(u_iw_i)  6n  4i  4; 2  i  n – 1

f^(u_n)  f(u_n1u_n)  f(u_nv_n)  f(u_nw_n)  5n  3

n

2

f^(v_2i1)  f(u

_{2i1 2i1})  f(v_2i1v_2i)  12i  8; 1  i 

v

f^(v_2i)  f(v_2i1v_2i)  f(v_2iu_2i)  12i  4; 1  i  ⁿ

2

n

2

f^(w_2i1)  f(u_2i1

w

_2i1)  f(w_2i1w_2i)  8n  12i  6; 1  i 

n

2

f^(w_2i)  f(w_{2i 1}w_2i)  f(w_2iu_2i)  8n  12i  2; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 8n  3}. Then the labels of

vertices are in the set

{4, 16, ..., 6n  8}  8, 20, ..., 6n  4  3n  1

 6n  4, 6n  8, ..., 2n  6  5n  3  8n  6, 8n  18, ..., 2n  6

 {8n  10, 8n  22, ..., 2n  2}. Here i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

f(uv) mod(8n  2) is injective.

_uvE(G)

Case 2: n  1 (mod 2)

Define edge labeling f : E(G)  {1, 3, 5, ...8n  9} is as follows:

f(u_iu_i1)  6n  2i  7; 1  i  n – 1

ⁿ

2



f(u_{2i 1 2i 1})  4i  3; 1  i 

v

ⁿ

2



f(u_2iv_2i)  4n  4i  7; 1  i 

ⁿ

2



f(v_iv_i1)  2n  4i  5; 1  i 

ⁿ

2



f(u_{2i 1}w_2i1)  4n  4i  5; 1  i 

144

VALANI DARSHANA K AND KANANI KAILAS K

ⁿ

2



f(u_2iw_2i)  4i  1; 1  i 

ⁿ

2



f(w_iw_{i 1})  2n  4i  3; 1  i 

The corresponding labels of vertices u_iand v_i

,

i  1, 2, 3... mod (8n  8)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  f(u₁w₁)  2n  3

;

f^(u_2i1)  f(u_2i2

u

_2i1)  f(u_2i1u_2i)  f(u

v

_{2i1 2i1})  f(u_2i1w_2i1

)

,

^{n 2}

2



 8n  16i  20; 2  i 

f^(u_2i)  f(u_{2i1 2i}

u )  f(u_2iu_2i1)  f(u_2iv_2i)  f(u_2iw_2i)

ⁿ

2



 8n  16i  16; 1  i 

f^(un)  f(u_n1u_n)  8n  9

ⁿ

2



f^(v_2i1)  f(u

_{2i 1 2i 1})  f(v_{2i 1}v_2i)  2n  8i  8; 1  i 

v

ⁿ

2



f^(v_2i)  f(u_2iv_2i)  f(v_{2i 1}v_2i)  6n  8i  12; 1  i 

ⁿ

2



f^(w_{2i 1})  f(u_{2i 1}

w_2i1)  f(w_2i1w_2i)  6n  8i  8; 1  i 

ⁿ

2



f^(w_2i)  f(u_2iw_2i)  f(w_{2i 1}w_2i)  2n  8i  4; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 8n  9} . Then the labels of

vertices are in the set {2n, 2n  8, ..., 6n  12}  6n  4, 6n  4, ..., 2n  8

 2n  3  8, 24, ..., 8n  16  {2n  3}  20, 36,, ..., 8n  20  8n  9

 {6n, 6n  8, ..., 2n  4}  2n  4, 2n  12, ..., 6n  8

.

Here

i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

f^(u) 

f(uv) mod (8n  8) is injective.

_uvE(G)

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

145

Example 2.4: The edge odd graceful labeling of the double alternate

quadrilateral snake DA(Q₆) and DA(Q₇) is shown in the following Figure 6

and 7.

Figure 6: The edge odd graceful labeling of double alternate quadrilateral

snake DA(Q6)

Figure 7: The edge odd graceful labeling of double alternate quadrilateral

snake DA(Q₇)

.

In each possibility the graph under consideration satisfies the

vertex conditions and edge conditions for an edge odd graceful labeling. Hence,

the double alternate quadrilateral snake DA(Q_n) is an edge odd graceful graph

for all n  2

.



Theorem 2.5: The alternate pentagonal snake A(PS_n) is an edge odd

graceful graph for all n  2

.

Proof: Let

from a path

to new vertices v_jand w_jrespectively and then joining v_jand w_jto the new

G

be a alternate pentagonal snake A(PS_n)which is obtained

P

with vertices u₁, u₂, ..., u_nby joining u_iand u_{i 1}(alternatively)

n

146

VALANI DARSHANA K AND KANANI KAILAS K

vertex

x

_j, where 1  i  n  1 for even n, 1  i  n  2 for odd

n

and

ⁿ

2



1  j 

.

Therefore,

ⁿ

2



V(G)  {u_i, v_j, w_j, x_j/1  i  n, 1  j 

}

ⁿ

2



ⁿ

2



E(G)  {u_iu_i1/1  i  n  1}  {u_2i1v_i/1  i 

}  {v x /1  i 

}

i i

ⁿ

2



ⁿ

2

 {x_iw_i/1  i 

}  {w u /1  i 

}

.

i

2i

Here note that

⁵ⁿ

,

if n  0 (mod 2)

 1, if n  1 (mod 2)



2

V(G) 



⁵ⁿ

2







3n  1, if n  0 (mod 2)



E(G) 



3n  3, if n  1 (mod 2)





Case 1: n  0 (mod 2)

Subcase 1: n  0 (mod 6)

Define edge labeling f : E(G)  {1, 3, 5, ..., 6n  2} is as follows:

f(u_iu_i1)  4n  2i  1; 1  i  n – 1

f(u_2i1v_i)  2n  4i  3; 1  i  ⁿ

2

n

2

f(v_ix_i)  4i  3; 1  i 

n

2

f(w_ix_i)  2n  4i  1; 1  i 

n

2

f(u_2iw_i)  4i  1; 1  i 

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

147

The corresponding labels of vertices u_iand v_i, i  1, 2, 3... mod(6n  2)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  6n  2

n2

2

f^(u_2i)  f(u_{2i1 2i}

u )  f(u_2iu_2i1)  f(u_2iw_i)  2n  12i  3; 1  i 

n

2

f^(u_2i1)  f(u_{2i1 2i}

u )  f(u_2iu_2i1)  f(u_2i1v_i)  4n  12i  9; 2  i 

f^(u_n)  f(u_{n 1}u_n)  f(w_{n /2}u_n)  8n  4

;

n

2

f^(v_i)  f(u_2i1v_i)  f(v_ix_i)  2n  8i  6; 1  i 

n

2

f^(x_i)  f(v_ix_i)  f(x_iw_i)  2n  8i  4; 1  i 

n

2

f^(w_i)  f(x_iw_i)  f(w_iu_2i)  2n  8i  2; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 6n  3} .Then the labels of

are in the set

{6n  2}  {2n  9, 2n  21, ..., 2n  13}

 {4n  15, 4n  27, ..., 4n  7}  2n  2  2n  2, 2n  10, ..., 6n  6

 {2n  4, 2n  12, ..., 6n  4}  2n  6, 2n  14, ..., 6n  2

vertices

.

Here

i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

f(uv) mod (6n  2) is injective.

_uvE(G)

Subcase 2: n  2, 4 (mod 6)

Define edge labeling f : E(G)  {1, 3, 5, ..., 6n  2} is as follows:

f(u_iu_i1)  4n  2i  1; 1  i  n – 1

n

2

f(u_{2i 1}v_i)  8i  7; 1  i 

n

2

f(v_ix_i)  8i  5; 1  i 

148

VALANI DARSHANA K AND KANANI KAILAS K

n

2

f(w_ix_i)  8i  3; 1  i 

n

2

f(u_2iw_i)  8i  1; 1  i 

The corresponding labels of vertices v_iand v_i, i  1, 2, 3... mod (6n  2)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  4n  2

;

n2

2

f^(u_2i)  f(u_2iu_2i1)  f(u_{2i1 2i}

u )  f(u_2iw_i)  2n  16i  3; 1  i 

n

2

f^(u_2i1)  f(u_{2i1 2i}

u )  f(u_2iu_2i1)  f(u_2i1v_i)  2n  16i  13; 2  i 

f^(u_n)  f(u_{n 1}u_n)  f(w_n/2u_n)  4n  2

;

n

2

f^(v_i)  f(u_2i1v_i)  f(v_ix_i)  16i  12; 1  i 

f^(x_i)  f(v_ix_i)  f(x_iw_i)  16i  8; 1  i  ⁿ

2

n

2

f^(w_i)  f(x_iw_i)  f(w_iu_2i)  16i  4; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 6n  3} . Then the labels of

are in the set

{4n  2}  2n  13, 2n  29, ..., 4n  17

 {2n  19, 2n  35, ..., 4n  11}  {8, 24, ..., 2n  6}  {12, 28, ..., 2n  2}

vertices

.

Here

i  j, f(v_i)  f(v_j)

.

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

f(uv) mod (6n  2) is injective.

_uvE(G)

Case 2: n  1 (mod 2)

Here note that



2

ⁿ

V(G)  5

 1

E(G)  3n  3

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

Define edge labeling f : E(G)  {1, 3, 5, ...6n  7} is as follows:

f(u_iu_i1)  6n  2i  5; 1  i n – 1

149

ⁿ

2



f(u_{2i 1}v_i)  4i  3; 1  i 

ⁿ

2



f(v_ix_i)  2n  4i  5; 1  i 

ⁿ

2



f(w_ix_i)  4i  1; 1  i 

ⁿ

2



f(u_2iw_i)  2n  4i  3; 1  i 

The corresponding labels of vertices u_iand v_i, i  1, 2, 3... mod (6n  6)

are

f^(u₁)  f(u₁u₂)  f(u₁v₁)  6n  6

;

ⁿ

2



f^(u_2i)  f(u_2iu_{2i 1})  f(u_{2i 1 2i}

u )  f(u_2iw_i)  2n  4i  1; 1  i 

ⁿ

2



f^(u_{2i 1})  f(u_{2i1 2i}

u )  f(u_2iu_{2i 1})  f(u_{2i 1}v_i)  6n  4i  1; 2  i 

f^(u_n)  f(u_n1u_n)  4n  3

;

ⁿ

2



f^(v_i)  f(u_2i1v_i)  f(v_ix_i)  2n  8i  8; 1  i 

ⁿ

2



f^(x_i)  f(v_ix_i)  f(x_iw_i)  2n  8i  6; 1  i 

ⁿ

2



f^(w_i)  f(x_iw_i)  f(w_iu_2i)  2n  8i  4; 1  i 

The labels of edges are in the set {1, 3, 5, ..., 6n  7}

.

Then the labels of vertices are in the set {6n  6}  2n  3, 2n  7,..., 6n  3

 {6n  9, 6n  13, ..., 4n  1}  4n  3  2n, 2n  8, ..., 6n  12

 {2n  2, 2n  10, ..., 6n  10}  2n  4, 2n  12, ..., 6n  8

.

Here i  j, f(v_i)  f(v_j)

.

150

VALANI DARSHANA K AND KANANI KAILAS K

Therefore, the induced function f^: V(G)  {0, 1, 2, ..., 2q  1}, defined

as f^(u) 

f(uv) mod (6n  6) is injective.

_uvE(G)

Example 2.6: The edge odd graceful labeling of the alternate pentagonal

snake A(PS₆) A(PS₁₀) and A(PS₉)is shown in Figure 8, 9 and 10.

,

Figure 8: The edge odd graceful labeling of alternate pentagonal snake A(PS₆)

.

Figure 9: The edge odd graceful labeling of alternate pentagonal snake A(PS₁₀)

.

Figure 10: The edge odd graceful labeling of alternate pentagonal snake

graph A(PS₉)

EDGE ODD GRACEFUL LABELING OF SOME SNAKE GRAPHS

151

In each possibility the graph under consideration satisfies the

vertex conditions and edge conditions for an edge odd graceful labeling.

Hence, the alternate pentagonal snake A(PS_n) is an edge odd graceful graph for

all n  2

3 Conclusion

In this paper, it is proved that double alternate triangular snake DA(T_n)

.



,

double alternate quadrilateral snake DA(Q_n) and alternate pentagonal snake

A(PS_n) are edge odd graceful graphs. To derive new families of graphs that admit

edge odd graceful labeling is an open area of research.

REFERENCES

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Combinatorics, Vol. 26, DS6X.

[3] Gnanajothi, R. (1991): Topics in graph theory, Ph. D. Thesis, Madurai Kamaraj

University.

[4] Golomb, S. W. (1972): How to number a graph, Graph theory and computing, Academic

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[5] Gross, J. and Yellen, J. (2003): Handbook of graph theory, CRC Press.

[6] Rosa A.

(1966): On certain valuations of the vertices of a graph, Theory of Graphs,

Rome, pp. 349-355.

[7] Seoud, M. and Salim, M. (2016): Further results on edge-odd graceful graphs, Turkish

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152

VALANI DARSHANA K AND KANANI KAILAS K

1. Research Scholar,

(Received, September 9, 2024)

Gujarat Technological University,

Ahmedabad , Gujarat, INDIA,

E-mail: darshanaalaiya@gmail.com

(Revised, October 4, 2024)

2. Government Engineering College,

Rajkot, Gujarat, INDIA.

E-mail: kananikkk@yahoo.co.in