http://www.scirp.org/journal/ojdm ISSN Online: 2161-7643 ISSN Print: 2161-7635

**Edge Product Cordial Labeling of Some Cycle **

**Related Graphs **

**Udayan M. Prajapati1, Nittal B. Patel2 **

1_{St. Xavier’s College, Ahmedabad, India }

2_{Shankersinh Vaghela Bapu Institute of Technology, Gandhinagar, India }

**Abstract **

For a graph *G*=

### (

*V G*

### ( ) ( )

,*E G*

### )

having no isolated vertex, a function### ( ) { }

: ,1

*f* *E G* → 0 is called an edge product cordial labeling of graph G, if the

induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.

**Keywords **

Graph Labeling, Edge Product Cordial Labeling

**1. Introduction **

We begin with simple, finite, undirected graph *G*=

### (

*V G*

### ( ) ( )

,*E G*

### )

having no isolatedvertex where *V G*

### ( )

and*E G*

### ( )

denote the vertex set and the edge set respectively,### ( )

*v G* and *E G*

### ( )

denote the number of vertices and edges respectively. For allother terminology, we follow Gross [1]. In the present investigations, *Cn* denotes

cycle graph with n vertices. We will give brief summary of definitions which are useful for the present work.

Definition 1. A graph labeling is an assignment of integers to the vertices or edges or both subject to the certain conditions. If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex (or an edge) labeling.

For an extensive survey on graph labeling and bibliography references, we refer to Gallian [2].

How to cite this paper: Prajapati, U.M. and Patel, N.B. (2016) Edge Product Cordial Labeling of Some Cycle Related Graphs.

Open Journal of Discrete Mathematics, 6, 268-278.

http://dx.doi.org/10.4236/ojdm.2016.64023

Received: June 8, 2016 Accepted: September 6, 2016 Published: September 9, 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

Definition 2. For a graph G, the edge labeling function is defined as *f* :*E G*

### ( ) { }

→ 0,1and induced vertex labeling function *

### ( ) { }

: 0,1

*f* *V G* → is given as if *e e*_{1}, _{2},,*en* are

all the edges incident to the vertex v then *

### ( )

### ( ) ( )

### ( )

1 2*n*

*f*

*v*=

*f e*

*f e*

*f e*.

Let *vf*

### ( )

*i*be the number of vertices of G having label i under *

*f* and *e _{f}*

### ( )

*i*be

the number of edges of G having label i under f for *i*=1, 2.

f is called an edge product cordial labeling of graph G if *v _{f}*

### ( )

0 −*v*

_{f}### ( )

1 ≤1 and### ( )

0### ( )

1 1*f* *f*

*e* −*e* ≤ . A graph G is called edge product cordial if it admits an edge product

cordial labeling.

Vaidya and Barasara [3] introduced the concept of the edge product cordial labeling as an edge analogue of the product cordial labeling.

Definition 3. The wheel *Wn*

### (

*n*>4

### )

is the graph obtained by adding a new vertexjoining to each of the vertices of *Cn*. The new vertex is called the apex vertex and the

vertices corresponding to *Cn* are called rim vertices of *Wn*. The edges joining rim

vertices are called rim edges.

Definition 4. The helm *Hn* is the graph obtained from a wheel *Wn* by attaching a

pendant edge to each of the rim vertices.

Definition 5. The closed helm *CHn* is the graph obtained from a helm *Hn* by

joining each pendant vertex to form a cycle. The cycle obtained in this manner is called an outer cycle.

Definition 6. The web *Wbn* is the graph obtained by joining the pendant vertices of

a helm *Hn* to form a cycle and then adding a pendant edge to each of the vertices of

the outer cycle.

Definition 7. The Closed Web graph *CWbn* is the graph obtained from a web graph
*n*

*Wb* by joining each of the outer pendent vertices consecutively to form a cycle.

Definition 8. [4] The Sunflower graph *SFn* is the graph obtained by taking a wheel

with the apex vertex *v*0 and the consecutive rim vertices *v v*1, 2,,*vn* and additional

vertices *w w*1, 2,,*wn* where *wi* is joined by edges to *vi* and *vi*+1

### (

mod*n*

### )

.Definition 9. [4] The Lotus inside a circle *LCn* is a graph obtained from a cycle
1 2 1

:

*n* *n*

*C* *u u* *u u* and a star graph *K*1,*n* with center vertex *v*0 and end vertices
1, 2, , *n*

*v v* *v* by joining each *vi* to *ui* and *ui*+1

### (

mod*n*

### )

.Definition 10. [5] Duplication of a vertex v of a graph G produces a new graph *G*′

by adding a new vertex *v*′ such that *N v*

### ( )

′ =*N v*

### ( )

. In other words,*v*′ is said to be

a duplication of v if all the vertices which are adjacent to v in G are also adjacent to *v*′

in *G*′.

Definition 11. [5] Duplication of a vertex *vk* by a new edge *e*=*v vk k*′ ′′ in a graph G

produces a new graph *G*′ such that *N v*

### ( ) {

*′ =*

_{k}*v v*,

_{k}*′′*

_{k}### }

and*N v*

### ( ) {

*′′ =*

_{k}*v v*,

_{k}*′*

_{k}### }

. Definition 12. The Flower graph*Fln*is the graph obtained from a helm

*Hn*by

joining each pendant vertex to the apex of the helm *Hn*.

**2. Main Result **

Theorem 1. Closed web graph *CWbn* is not an edge product cordial graph.

of *Wn*. Let *ei* =*v vi i*+1, for each *i*=1, 2,,*n* (here subscripts are modulo n). Let
0

*i* *i*

*f* =*v v* . Let 1, 2, ,

*a* *a* *a*

*n*

*v v* *v* be the new vertices corresponding to *v v v*_{1}, _{2}, _{3},,*vn*

respectively. Form the edge *a*
*i*

*f* by joining *vi* and
*a*
*i*

*v* . Form the cycle by creating

edges 1

### (

1 2### ) (

, 2 2 3### )

, ,### (

1### )

*a* *a* *a* *a* *a* *a* *a* *a* *a*

*n* *n*

*e* =*v v* *e* =*v v* *e* =*v v* . The resulting graph is called a helm

graph. Let 1, 2, ,

*b* *b* *b*

*n*

*v v* *v* be the new vertices corresponding to *v v*_{1}*a*, _{2}*a*,,*v _{n}a* respec-

tively. Form the edge *c*
*i*

*f* by joining *a*
*i*

*v* and *b*
*i*

*v* . Form the cycle by creating the edges

### (

### ) (

### )

### (

### )

1 1 2 , 2 2 3 , , 1

*v* *b b* *b* *b b* *b* *b b*

*n* *n*

*e* =*v v* *e* =*v v* *e* =*v v* . The resulting graph is a closed web graph

*n*

*CWb* . Thus *E CWb*

### (

_{n}### )

=6*n*and

*V CWb*

### (

_{n}### )

=3*n*+1. We are trying to define the mapping

*f*:

*E CWb*

### (

_{n}### ) { }

→ 0,1 we consider following two cases:Case 1: If n is odd then in order to satisfy the edge condition for edge product cordial graph it is essential to assign label 0 to 3n edges out of 6n edges. So in this context, the edges with label 0 will give rise at least 3 1

2

*n*

_{ +}

vertices with label 0 and at most

3 2

*n*

vertices with label 1 out of 3*n*+1 vertices. Therefore *vf*

### ( )

0 −*vf*

### ( )

1 ≥2. So*n*

*CWb* is not an edge product cordial graph for odd n.

Case 2: If n is even then in order to satisfy the edge condition for edge product cordial graph it is essential to assign label 0 to 3n edges out of 6n edges. So in this context, the edges with label 0 will give rise at least 3 2

2

*n*_{+} _{ vertices with label 0 and at }

most 3 1

2
*n*_{−}

vertices with label 1 out of 3*n*+1 vertices.

From both the cases *v _{f}*

### ( )

0 −*v*

_{f}### ( )

1 ≥2, so*CWb*is not an edge product cordial

_{n}graph.

Theorem 2. Lotus inside circle *LCn* is not an edge product cordial graph.

Proof. Let *v*0 be the apex vertex and *v v v*1, 2, 3,,*vn* be the consecutive vertices of

star graph *K*1,*n*. Let *ei* =*v v*0 *i* for each *i*=1, 2,,*n*. Let 1, 2, ,

*a* *a* *a*

*n*

*v v* *v* be the new

vertices corresponding to *v v v*1, 2, 3,,*vn* respectively. Form the cycle by creating the

edges 1

### (

1 2### ) (

, 2 2 3### )

, ,### (

1### )

*a* *a* *a* *a* *a* *a* *a* *a* *a*

*n* *n*

*e* =*v v* *e* =*v v* *e* =*v v* . Let *e _{i}l* =

*v v*and

_{i i}a*r*

*a*

_{1}

*i*

*i i*

*e* =*v v*_{+} . The re-

sulting graph is Lotus inside circle *Lcn*. Thus *E LC*

### (

*n*

### )

=4*n*and

*V LC*

### (

*n*

### )

=2*n*+1.

We are trying to define the mapping *f* :*E LC*

### (

_{n}### ) { }

→ 0,1 . In order to satisfy an edge condition for edge product cordial graph it is essential to assign label 0 to 2n edges out of 4n edges. So in this context, the edges with label 0 will give rise to at least*n*+2

vertices with label 0 and at most *n*−1 vertices with label 1 out of 2*n*+1 vertices.

Therefore *v _{f}*

### ( )

0 −*v*

_{f}### ( )

1 ≥3.So, *LCn* is not an edge product cordial graph.

Theorem 3. Sunflower graph *SFn* is an edge product cordial graph for *n*≥3.

Proof. Let *SFn* be the sunflower graph, where *v*0 is the apex vertex, *v v*1, 2,,*vn*

be the consecutive rim vertices of *Wn* and *w w*1, 2,,*wn* be the additional vertices

where *wi* is joined to *vi* and *vi*+1

### (

mod*n*

### )

. Let*e e e*1, 2, ,3 ,

*en*be the consecutive

rim edges of the *Wn*. *e e*1′ ′, 2,,*en*′ are the corresponding edges joining apex vertex *v*0

to the vertices *v v*1, 2,,*vn* of the cycle. Let for each 1≤ ≤*i* *n*,
*l*
*i*

*e* be the edges joining

*i*

*w* to *vi* and
*r*
*i*

### (

_{n}### )

2 1*V SF* = *n*+ . Define the mapping *f* :*E SF*

### (

*n*

### ) { }

→ 0,1 as follows:### ( )

0 if### {

_{{ }}

, ### }

, 1, 2, 3, , ;1 if , , 1, 2, 3, , .

*i* *i*
*l* *r*
*i* *i*

*e* *e e* *i* *n*
*f e*

*e* *e e* *i* *n*

′

∈ ∀ =

= _{∈} _{∀ =}

In view of the above defined labeling pattern we have, *vf*

### ( ) {

0 =*v v v*0, ,1 2,,

*vn*

### }

and### ( ) {

1 1, 2, ,### }

*f* *n*

*v* = *w w* *w* . So, *v _{f}*

### ( )

0 =*v*

_{f}### ( )

1 + = +1*n*1 and

*e*

_{f}### ( )

0 =*e*

_{f}### ( )

1 =2*n*. Thus

### ( )

0### ( )

1 1*f* *f*

*v* −*v* ≤ and *e _{f}*

### ( )

0 −*e*

_{f}### ( )

1 ≤1. Thus f admits an edge product cordial on*n*

*SF* . Hence, *SFn* is an edge product cordial graph.

Illustration 1. Graph *SF*6 and its edge product cordial labeling is shown in Figure

1.

Theorem 4. The graph obtained from duplication of each of the vertices *wi* for
1, 2, ,

*i*= *n* by a new vertex in the sunflower graph *SFn* is an edge product cordial

graph if and only if n is even.

Proof. Let *SFn* be sunflower graph, where *v*0 is the apex vertex, *v v*1, 2,,*vn* be

the consecutive rim vertices of *Wn* and *w w*1, 2,,*wn* be the additional vertices where
*i*

*w* is joined to *vi* and *vi*+1

### (

mod*n*

### )

. Let*e e e*1, 2, ,3 ,

*en*be the consecutive rim edges

of the *Wn*. *e e*1′ ′, 2,,*en*′ are the corresponding edges joining the apex vertex *v*0 to the

vertices *v v*1, 2,,*vn* of the cycle. Let for each 1≤ ≤*i* *n*,
*l*
*i*

*e* be the edges joining *wi*

to *vi* and
*r*
*i*

*e* be the edges joining *w _{i}* to

*v*+1 mod

_{i}### (

*n*

### )

. Let G be the graph obtainedfrom *SFn* by duplication of the vertices *w w*1, 2,,*wn* by the new vertices
1, 2, , *n*

*u u* *u* respectively. Let for each 1≤ ≤*i* *n*, *l*
*i*

*f* be the edges joining *u _{i}* to

*vi*

and *r*

*i*

*f* be the edges joining *ui* to *vi*+1

### (

mod*n*

### )

. Thus*E G*

### ( )

=6*n*and

### ( )

3 1*V G* = *n*+ . We consider the following two cases:

Case 1: If n is odd, define the mapping *g E G*:

### ( ) { }

→ 0,1 in order to satisfy edge condition for edge product cordial graph it is essential to assign label 0 to 3n edges out of 6n edges. So in this context, the edges with label 0 will give rise at least 3 12

*n*

_{ +}

vertices with label 0 and at most 3 2

*n*

vertices with label 1 out of 3*n*+1 vertices.

Therefore *v _{g}*

### ( )

0 −*v*

_{g}### ( )

1 ≥2 . So the duplication of vertex*w*by vertex

_{i}*ui*in

sunflower graph is not edge product cordial for odd n.

Case 2: If n is even, define the mapping *g E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### { }

### { }

### {

### }

0 if , , 1, 2, 3, , ;

1 if , , 1, 2, 3, , ;

2

0 if , , 1, 2, , ;

2 2

1 if , , 1, 2, 3, , .

*i* *i*

*l* *r*
*i* *i*

*l* *r*
*i* *i*

*l* *r*
*i* *i*

*e* *e e* *i* *n*
*n*
*e* *e e* *i*

*g e*

*n* *n*

*e* *e e* *i* *n*
*e* *f* *f* *i* *n*

′

∈ ∀ =

_{∈} _{∀ =}

=

∈ ∀ = + +

_{∈} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 21 2 2 2

0 , , , , , , , ,

*g* *n* *n* *n* *n*

*v* *v v v* *v w* *w* *w*

+ +

=

and

### ( )

1 2 1 2 21 , , , , , , ,

*g* *n* *n*

*v* = *u u* *u w w* *w* _{}

.

So

### ( )

0### ( )

1 1 3 12

*g* *g*

*n*

*v* =*v* + = + and *e _{g}*

### ( )

0 =*e*

_{g}### ( )

1 =3*n*. Thus

*vg*

### ( )

0 −*vg*

### ( )

1 ≤1 and### ( )

0### ( )

1 1*g* *g*

*e* −*e* ≤ . Thus g admits an edge product cordial labeling on G. So, G is an

edge product cordial for even n.

Illustration 2. Graph G obtained from *SF*6 by duplication of each of the vertices
1, 2, 3, 4, 5, 6

*w w w w w w* by new vertices *u u u u u u*1, 2, 3, 4, 5, 6 and its edge product cordial

labeling is shown in Figure 2.

Theorem 5. The graph obtained from duplication of each of the vertices *wi* for
1, 2, ,

*i*= *n* by a new edges *fi* in the sunflower graph *SFn* is an edge product

cordial graph.

Proof. Let *SFn* be sunflower graph, where *v*0 is the apex vertex, *v v*1, 2,,*vn* be

the consecutive rim vertices of *Wn* and *w w*1, 2,,*wn* be the additional vertices where

*i*

*w*

is joined to *vi*and

*vi*+1

### (

mod*n*

### )

. Let*e e e*1, 2, ,3,

*en*be the consecutive rim

edges of *Wn*. *e e*1′ ′, 2,,*en*′ are the corresponding edges joining apex vertex *v*0 to the

vertices *v v*1, 2,,*vn* of the cycle. Let for each 1≤ ≤*i* *n*,
*l*
*i*

*e* be the edges joining *w _{i}*

to *vi* and
*r*
*i*

*e* be the edges joining *w _{i}* to

*vi*+1 mod

### (

*n*

### )

. Let G be the graph obtained from*SFn*by duplication of the vertices

*w w*1, 2,,

*wn*by corresponding new edges

1, 2, , *n*

*f* *f* *f* with new vertices *l*
*i*

*u* and *uir* such that
*l*
*i*

*u* and *uir* join to the vertex

*i*

*w*. Let for each 1≤ ≤*i* *n*, *fi* be the edges joining the vertices
*l*
*i*

*u* and *uir* and for

each 1≤ ≤*i* *n*, *l*
*i*

*f* be the edges joining *uil* to *wi* and
*r*
*i*

*f* be the edges joining *uir*

to *wi*. Thus *E G*

### ( )

=7*n*and

*V G*

### ( )

=4*n*+1. We consider the following two cases:

Case 1: If n is odd, define the mapping *g E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### {

### }

### {

1### }

1

1 if , 1, 2, 3, , ;

2 1

1 if , 2, 3, , ;

2

1

1 if , , , , , 1, 2, 3, , ;

2

1 3 5

0 if , , , , , , ;

2 2 2

0 if ;

3 5 7

0 if , , , , , , , , , .

2 2 2

*i*

*i*

*l* *r* *l* *r*

*i* *i* *i* *i* *i*

*i* *i*

*l* *r* *l* *r*

*i* *i* *i* *i* *i*
*n*
*e* *e* *i*

*n*
*e* *e* *i*

*n*
*e* *e e* *f u u* *i*

*g e*

*n* *n* *n*

*e* *e e* *i* *n*

*e* *e*

*n* *n* *n*

*e* *e e* *f u u* *i* *n*

−
_{=} _{∀ =}
_{−}
= ′∀ =
+
∈ ∀ =
=
_{∈} _{′} _{∀ =} + + +
_{=} _{′}
+ + +
_{∈} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 1 1 1 1 1 1 1 11 2 1 2 1 2 1 2

2 2 2 2 2 2 2 2

0 , , , , , , *l* , *l* , , *l*, *r* , *r* , , *r*, , , ,

*g* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n*

*v* *v v v*−_{+} *v*−_{+} *v u* +_{+} *u* +_{+} *u u*+_{+} *u* +_{+} *u w*+_{+} *w*+_{+} *w*

=

and

### ( )

2 3 1 1 2 1 1 2 1 1 2 12 2 2 2

1 , , , , *l*, *l*, , *l* , *r*, *r*, , *r* , , , , .

*g* *n* *n* *n* *n*

*v* = *v v* *v*_{−} *u u* *u*_{+} *u u* *u*_{+} *w w* *w*_{+} _{}

So,

### ( )

0### ( )

1 1 2 1*g* *g*

*v* =*v* + = *n*+ and

### ( )

1 72

*g*

*n*
*e* =

,

### ( )

7 0 2*g*

*n*

*e*=

.

Case 2: If n is even, define *g E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### { }

### {

### }

### {

### }

### { }

12

1 if , 1, 2, 3, , ;

2 2

1 if , 2, 3, 4, , ;

2

1 if , , 1, 2, 3, , ;

2 2

1 if , , , 1, 2, 3, , ;

2

2 4

0 if , , , , , , ;

2 2 2

0 if ;

2 4 6

0 if , , , , , , ;

2 2 2

0 if , ,

*i*

*i*

*l* *r*
*i* *i*

*l* *r*
*i* *i* *i*

*i* *i*
*l* *r*
*i* *i*
*l*
*i* *i*
*n*

*e* *e* *i*

*n*

*e* *e* *i*

*n*

*e* *e e* *i*

*n*

*e* *f u u* *i*

*g e*

*n n* *n*

*e* *e e* *i* *n*

*e* *e*

*n* *n* *n*

*e* *e e* *i* *n*

*e* *f u u*

− = ∀ = − ′ = ∀ = ∈ ∀ = + ∈ ∀ = = + + ′ ∈ ∀ = ′ = + + + ∈ ∀ = ∈

### {

### }

4 6 8, , , , , .

2 2 2

*r*
*i*

*n* *n* *n*

In view of the above defined labeling pattern we have,

### ( )

0 1 2 2 2 2 2 21 2 1 2 1 2 1 2

2 2 2 2 2 2 2 2

0 , , , , , , *l* , *l* , , *l*, *r* , *r* , , *r*, , , ,

*g* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n* *n*

*v* *v v v* −_{+} *v*−_{+} *v u*+ _{+} *u*+ _{+} *u u*+ _{+} *u*+ _{+} *u w* _{+} *w* _{+} *w*

=

and

### ( )

2 3 2 1 2 2 1 2 2 1 22 2 2 2

1 , , , , *l*, *l*, , *l* , *r*, *r*, , *r* , , , , .

*g* *n* *n* *n* *n*

*v* = *v v* *v*_{−} *u u* *u* _{+} *u u* *u* _{+} *w w* *w* _{}

So,

### ( )

0### ( )

1 1 2 1*g* *g*

*v* =*v* + = *n*+ and

### ( )

0### ( )

1 7 .2

*g* *g*

*n*
*e* =*e* =

From both the cases *v _{g}*

### ( )

0 −*v*

_{g}### ( )

1 ≤1 and*e*

_{g}### ( )

0 −*e*

_{g}### ( )

1 ≤1. Thus, g admits an edge product cordial labeling on G. So the graph G is an edge product cordial graph.Illustration 3. Graph G obtained from *SF*5 by duplication of each of the vertices
1, 2, 3, 4, 5

*w w w w w* by corresponding new edges *f*1,*f*2,*f*3,*f*4,*f*5 and its edge product

cordial labeling is shown in Figure 3.

Theorem 6. The graph obtained by duplication of each of the vertices in the
sunflower graph *SFn* is not an edge product cordial graph.

Proof. Let *SFn* be the sunflower graph, where *v*0 is the apex vertex, *v v*1, 2,,*vn*

be the consecutive vertices of the cycle *Cn* and *w w*1, 2,,*wn* be the additional

vertices where *wi* is joined to *vi* and *vi*+1

### (

mod*n*

### )

. Let*e e e*1, 2, ,3 ,

*en*be the

consecutive edges of the cycle *Cn*. *e e*1′ ′, 2,,*en*′ are the corresponding edges joining

the apex vertex *v*0 to the vertices *v v*1, 2,,*vn* of the cycle. Let for each 1≤ ≤*i* *n*,
*l*
*i*

*e*

be the edge joining *wi* to *vi* and
*r*
*i*

*e* be the edge joining *w _{i}* to

*vi*+1

### (

mod*n*

### )

. Let Gbe the graph obtained from *SFn* by duplication of each of the vertices
0, ,1 2, , *n*, 1, 2, , *n*

*v v v* *v w w* *w* by the vertices *v v v*′ ′ ′0, ,1 2,,*v w wn*′ ′ ′, 1, 2,,*wn*′ respectively.

Let 1, 2, , 2

*a* *a* *a*

*n*

*e e* *e* be the edges joining the vertex *w _{i}*′ to the adjacent vertex of

*wi*for 1, 2, ,

*i*= *n*. *e e*_{1}*b*, _{2}*b*,,*e*_{5}*b _{n}* are the edges joining the vertex

*vi*′ to the adjacent vertex

of *vi* for *i*=1, 2,,*n* and 1, 2, ,

*c* *c* *c*

*n*

*e e* *e* are the edges joining the vertex *v*0′ to the

adjacent vertex of c in the sunflower graph. Thus *E G*

### ( )

=12*n*and

*V G*

### ( )

=4*n*+2. We are trying to define

*f*:

*E G*

### ( ) { }

→ 0,1 .In order to satisfy the edge condition for edge product cordial graph, it is essential to assign label 0 to 6n edges out of 12n edges. So in this context, the edge with label 0 will give rise at least 4 1 1

2

*n*+

_{ +}

vertices with label 0 and at most

4 1

2

*n*+

vertices with

label 1 out of 4*n*+2 vertices. Therefore, *v _{f}*

### ( )

0 −*v*

_{f}### ( )

1 ≥2. So the graph G is not an edge product cordial graph.Theorem 7. The graph obtained by subdividing the edges *w vi i* and *w vi i*+1

(subscripts are mod n) for all i = 1,2,...,n by a vertex in the sunflower graph *SFn* is an

edge product cordial.

Proof. Let *SFn* be the sunflower graph, where *v*0 is the apex vertex, *v v*1, 2,,*vn*

be the consecutive rim vertices of *Wn* and *w w*1, 2,,*wn* be the additional vertices

where *wi* is joined to *vi* and *vi*+1

### (

mod*n*

### )

. Let*e e e*1, 2, ,3 ,

*en*be the consecutive

rim edges of *wn*. *e e*1′ ′, 2,,*en*′ are the corresponding edges joining apex vertex *v*0 to

the vertices *v v*1, 2,,*vn* of the cycle. Let G be the graph obtained from *SFn* by

subdividing the edges *w vi i* and *w vi i*+1

### (

mod*n*

### )

for all*i*=1, 2,,

*n*by a vertex

*l*

*i*

*u*

and *r*
*i*

*u* respectively. Let for each 1≤ ≤*i* *n*, *l*
*i*

*e* be the edges joining *u _{i}l* to

*vi*and

*r*

*i*

*e* be the edges joining *u _{i}r* to

*v*

_{i}_{+}

_{1}

### (

mod*n*

### )

. Let for each 1≤ ≤*i*

*n*,

*l*

*i*

*f* be the edges

joining *wi* to
*l*
*i*

*u* and *f _{i}r* be the edges joining

*wi*to

*r*

*i*

*u* . Thus, *E G*

### ( )

=6*n*and

### ( )

4 1*V G* = *n*+ . We consider the following two cases:

Case 1: If n is odd, define *g E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### {

### }

0 if , , = 1, 2, 3, , ;

3 5 7

0 if , , , , , ;

2 2 2

1 3 5

0 if , , , , , ;

2 2 2

1

1 if , 1, 2, 3, , ;

2 1

1 if , 1, 2, 3, , ;

2

1 if , , 1, 2, 3, , .

*i* *i*
*l*
*i*
*r*
*i*
*l*
*i*
*r*
*i*
*l* *r*
*i* *i*

*e* *e e* *i* *n*

*n* *n* *n*

*e* *e* *i* *n*

*n* *n* *n*

*e* *e* *i* *n*

*g e*

*n*

*e* *e* *i*

*n*

*e* *e* *i*

*e* *f* *f* *i* *n*

′
∈ ∀
+ + +
_{=} _{∀ =}
_{+} _{+} _{+}
_{=} _{∀ =}
= _{+}
= ∀ =
_{−}
= ∀ =
_{∈} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 2 1 1 1 11 2 1 2 2 2 2 2

0 , , , , , *l* , *l* , , *l*, *r* , *r* , , *r*

*g* *n* *n* *n* *n* *n* *n* *n*

*v* *v v v* *v u*+_{+} *u* +_{+} *u u* −_{+} *u*−_{+} *u*

=

and

### ( )

1 2 1 1 2 1 1 22 2

1 *l*, *l*, , *l* , *r*, *r*, , *r* , , ,

*g* *n* *n* *n*

*v* *u u* *u* + *u u* *u*−*w w* *w*

=

. So *vg*

### ( )

0 =*vg*

### ( )

1 + =1 2*n*+1 and

### ( )

0### ( )

1 3*g* *g*

Case 2: If n is even, define *g E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### { }

### { }

### {

### }

0 if , , 1, 2, 3, , ;

2 4 6

0 if , , , , , , ;

2 2 2

1 if , , 1, 2, 3, , ;

2

1 if , , 1, 2, 3, , .

*i* *i*

*l* *r*
*i* *i*

*r* *l*
*i* *i*

*l* *r*
*i* *i*

*e* *e e* *i* *n*
*n* *n* *n*

*e* *e e* *i* *n*

*g e*

*n*
*e* *e e* *i*

*e* *f* *f* *i* *n*

′

∈ ∀ =

_{+} _{+} _{+}

_{∈} _{∀ =}

=

∈ ∀ =

_{∈} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 21 2 1 2
2 2 2 2
0 , , , , , *l* , *l* , , *l*, *r* , *r* , , *r*

*g* *n* *n* *n* *n* *n* *n* *n*

*v* *v v v* *v u* *u* *u u* *u* *u*

+ + + +

=

and

### ( )

1 2 1 2 1 22 2

1 *l*, *l*, , *l*, *r*, *r*, , *r* , , ,

*g* *n* *n* *n*

*v* = *u u* *u u u* *u w w* *w*

. S o *vg*

### ( )

0 =*vg*

### ( )

1 + =1 2*n*+1 a n d

### ( )

0### ( )

1 3*g* *g*

*e* =*e* = *n*.

From both the cases *vg*

### ( )

0 −*vg*

### ( )

1 ≤1 and*eg*

### ( )

0 −*eg*

### ( )

1 ≤1. Thus g admits anedge product cordial labeling on G. So the graph G is an edge product cordial graph.
Illustration 4. Graph G obtained from *SF*8 by subdividing the edges *w vi i* and

### (

### )

1 mod

*i i*

*w v*_{+} *n* for all *i*=1, 2,,8 by a vertex *l*
*i*

*u* and *uir* respectively for
1, 2, ,8

*i*= and its edge product cordial labeling is shown in Figure 4.

Theorem 8. The graph obtained by flower graph *Fln* by adding n pendant vertices

to the apex vertex *v*0 is an edge product cordial graph.

Proof. The Flower graph *Fln* is the graph obtained from a helm *Hn* by joining

each pendant vertex to the apex vertex of the helm *Hn*. Let G be the graph obtained

from the flower graph *Fln* by adding n pendant vertices *u u*1, 2,,*un* to the apex

vertex *v*0. Let *v v*1, 2,,*vn* be the rim vertices and *w w*1, 2,,*wn* be the pendant

vertices to *v v*1, 2,,*vn* respectively.

Let *e e e*1, 2, ,3 ,*en* be the consecutive rim edges of *Wn*. *e e*1′ ′, 2,,*en*′ are the cor-

responding edges joining the apex vertex *v*0 to the vertices *v v*1, 2,,*vn* of the cycle. Let
*a*

*i* *i i*

*e* =*w v* for each *i*=1, 2,,*n*. Let *eib*=*w vi* _{0} for each *i*=1, 2,,*n*. Let 0
*c*

*i* *i*

*e* =*u v*

for each *i*=1, 2,,*n*. Thus *E G*

### ( )

=5*n*and

*V G*

### ( )

=3*n*+1. We consider two cases:

Case 1: If n is even, define the mapping *f E G*:

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### {

### }

0 if , , 1, 2, 3, , ;

1 if , , 1, 2, 3, , ;

1 if , 1, 2, 3, , ;

2

2 4 6

0 if , , , , , .

2 2 2

*i* *i*
*a* *b*
*i* *i*
*c*
*i*
*c*
*i*

*e* *e e* *i* *n*
*e* *e e* *i* *n*

*n*
*f e*

*e* *e* *i*

*n* *n* *n*

*e* *e* *i* *n*

′
∈ ∀ =
∈ ∀ =
= _{=} _{∀ =}
_{+} _{+} _{+}
_{=} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 22 2

0 , , , , , 1, 2, ,

*f* *n* *n* *n* *n*

*v* =*v v v* *v u* + *u* + *u*

and

### ( )

1 2 1 22

1 , , , , , , , .

*f* *n* *n*

*v* = *w w* *w u u* *u*

So

### ( )

### ( )

3

0 1 1 1

2

*f* *f*

*n*

*v* =*v* + = + and

### ( )

### ( )

50 1 .

2

*f* *f*

*n*
*e* =*e* =

Case 2: If n is odd, define the mapping *f* :*E G*

### ( ) { }

→ 0,1 as follows:### ( )

### {

### }

### {

### }

0 if , , 1, 2, 3, , ;

1 if , , 1, 2, 3, , ;

1

= _{1} _{if} _{,} _{1, 2, 3,} _{,} _{;}

2

3 5 7

0 if , , , , , .

2 2 2

*i* *i*
*a* *b*
*i* *i*
*c*
*i*
*c*
*i*

*e* *e e* *i* *n*
*e* *e e* *i* *n*

*n*
*f e* _{e}_{e}_{i}

*n* *n* *n*

*e* *e* *i* *n*

′
∈ ∀ =
∈ ∀ =
_{+}
_{=} _{∀ =}
_{+} _{+} _{+}
_{=} _{∀ =}

In view of the above defined labeling pattern we have,

### ( )

0 1 2 1 11 2 2 2

0 , , , , , , , ,

*f* *n* *n* *n* *n*

*v* *v v v* *v u* +_{+} *u* +_{+} *u*

=

and

### ( )

1 2 1 2 12

1 , , , , , , , .

*f* *n* *n*

*v* *w w* *w u u* *u* +

=

So

### ( )

### ( )

3 1

0 1

2

*f* *f*

*n*

*v* =*v* = + and

### ( )

1 52

*f*

*n*
*e* =

,

### ( )

50 .

2
*f*

*n*
*e* =

From both the cases *v _{f}*

### ( )

0 −*v*

_{f}### ( )

1 ≤1 and*e*

_{f}### ( )

0 −*e*

_{f}### ( )

1 ≤1. Thus, f admits an edge product cordial labeling on G. So G is an edge product cordial graph.Illustration 5. Graph G obtained from *Fl*5 by adding 5 pendent vertices
1, 2, 3, 4, 5

*u u u u u* to the apex vertex *v*0 and its edge product cordial labeling is shown in

Figure 5. G and its edge product cordial labeling.

**3. Concluding Remarks **

We investigated eight results on the edge product cordial labeling of various graph generated by a cycle. Similar problem can be discussed for other graph families.

**Acknowledgements **

The authors are highly thankful to the anonymous referee for valuable comments and constructive suggestions. The first author is thankful to the University Grant Commis-sion, India for supporting him with Minor Research Project under No. F. 47-903/14 (WRO) dated 11th March, 2015.

**References **

[1] Gross, J.L. and Yellen, J. (Eds.) (2004) Handbook of Graph Theory. CRC Press, Boca Raton. [2] Gallian, J.A. (2014) A Dynamic Survey of Graph Labeling. The Electronic Journal of

Com-binatorics, 17, #DS6. http://www.combinatorics.org

[3] Vaidya, S.K. and Barasara, C.M. (2012) Edge Product Cordial Labeling of Graphs. Journal

of Mathematical and computational Science, 2, 1436-1450.

[4] Ponraj, R., Sathish Narayanan, S. and Kala, R. (2015) A Note on Difference Cordial Graphs.

Palestine Journal of Mathematics, 4, 189-197.

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