Edge Product Cordial Labeling of Some Cycle Related Graphs

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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

1St. Xavier’s College, Ahmedabad, India

2Shankersinh 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 isolated

vertex 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 all

other 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/

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

( ) { }

→ 0,1

and induced vertex labeling function *

( ) { }

: 0,1

f V G → is given as if e e1, 2,,en are

all the edges incident to the vertex v then *

( )

( ) ( )

( )

1 2 n f v = f e f ef e .

Let vf

( )

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

f and ef

( )

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 vf

( )

0 −vf

( )

1 ≤1 and

( )

0

( )

1 1

f f

ee ≤ . 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 vertex

joining 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 v0 and the consecutive rim vertices v v1, 2,,vn and additional

vertices w w1, 2,,wn where wi is joined by edges to vi and vi+1

(

modn

)

.

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

:

n n

C u uu u and a star graph K1,n with center vertex v0 and end vertices 1, 2, , n

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

(

modn

)

.

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 vk, k′′

}

and N v

( ) {

k′′ = v vk, 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.

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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 vv be the new vertices corresponding to v v v1, 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 ve =v v . The resulting graph is called a helm

graph. Let 1, 2, ,

b b b

n

v vv be the new vertices corresponding to v v1a, 2a,,vna 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 ve =v v . The resulting graph is a closed web graph

n

CWb . Thus E CWb

(

n

)

=6n and V CWb

(

n

)

=3n+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 3n+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 3n+1 vertices.

From both the cases vf

( )

0 −vf

( )

1 ≥2, so CWbn is not an edge product cordial

graph.

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

Proof. Let v0 be the apex vertex and v v v1, 2, 3,,vn be the consecutive vertices of

star graph K1,n. Let ei =v v0 i for each i=1, 2,,n. Let 1, 2, ,

a a a

n

v vv be the new

vertices corresponding to v v v1, 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 ve =v v . Let eil =v vi ia and r a1 i i i

e =v v+ . The re-

sulting graph is Lotus inside circle Lcn. Thus E LC

(

n

)

=4n and V LC

(

n

)

=2n+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 2n+1 vertices.

Therefore vf

( )

0 −vf

( )

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 v0 is the apex vertex, v v1, 2,,vn

be the consecutive rim vertices of Wn and w w1, 2,,wn be the additional vertices

where wi is joined to vi and vi+1

(

modn

)

. Let e e e1, 2, ,3 ,en be the consecutive

rim edges of the Wn. e e1′ ′, 2,,en′ are the corresponding edges joining apex vertex v0

to the vertices v v1, 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

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(

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 v0, ,1 2,,vn

}

and

( ) {

1 1, 2, ,

}

f n

v = w ww . So, vf

( )

0 =vf

( )

1 + = +1 n 1 and ef

( )

0 =ef

( )

1 =2n. Thus

( )

0

( )

1 1

f f

vv ≤ and ef

( )

0 −ef

( )

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

n

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

Illustration 1. Graph SF6 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 v0 is the apex vertex, v v1, 2,,vn be

the consecutive rim vertices of Wn and w w1, 2,,wn be the additional vertices where i

w is joined to vi and vi+1

(

modn

)

. Let e e e1, 2, ,3 ,en be the consecutive rim edges

of the Wn. e e1′ ′, 2,,en′ are the corresponding edges joining the apex vertex v0 to the

vertices v v1, 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 wi to vi +1 mod

(

n

)

. Let G be the graph obtained

from SFn by duplication of the vertices w w1, 2,,wn by the new vertices 1, 2, , n

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

f be the edges joining ui to vi

and r

i

f be the edges joining ui to vi+1

(

modn

)

. Thus E G

( )

=6n 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 1

2

n

 +    

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vertices with label 0 and at most 3 2

n

   

  vertices with label 1 out of 3n+1 vertices.

Therefore vg

( )

0 −vg

( )

1 ≥2 . So the duplication of vertex wi by vertex 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 2

1 2 2 2

0 , , , , , , , ,

g n n n n

v v v v v w w w

+ +

 

 

=  

 

    and

( )

1 2 1 2 2

1 , , , , , , ,

g n n

v = u u u w w w 

 

    .

So

( )

0

( )

1 1 3 1

2

g g

n

v =v + = + and eg

( )

0 =eg

( )

1 =3n. Thus vg

( )

0 −vg

( )

1 ≤1 and

( )

0

( )

1 1

g g

ee ≤ . 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 SF6 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 u1, 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.

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Proof. Let SFn be sunflower graph, where v0 is the apex vertex, v v1, 2,,vn be

the consecutive rim vertices of Wn and w w1, 2,,wn be the additional vertices where

i

w

is joined to vi and vi+1

(

modn

)

. Let e e e1, 2, ,3,en be the consecutive rim

edges of Wn. e e1′ ′, 2,,en′ are the corresponding edges joining apex vertex v0 to the

vertices v v1, 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 wi to vi +1 mod

(

n

)

. Let G be the graph obtained from SFn by duplication of the vertices w w1, 2,,wn by corresponding new edges

1, 2, , n

f ff 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

( )

=7n and V G

( )

=4n+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 1

1 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 1

2 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 7

2

g

n e =   

 ,

( )

7 0 2 g n e =   

 .

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

( ) { }

→ 0,1 as follows:

( )

{ }

{

}

{

}

{ }

1

2

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

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In view of the above defined labeling pattern we have,

( )

0 1 2 2 2 2 2 2

1 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 2

2 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 vg

( )

0 −vg

( )

1 ≤1 and eg

( )

0 −eg

( )

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 SF5 by duplication of each of the vertices 1, 2, 3, 4, 5

w w w w w by corresponding new edges f1,f2,f3,f4,f5 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 v0 is the apex vertex, v v1, 2,,vn

be the consecutive vertices of the cycle Cn and w w1, 2,,wn be the additional

vertices where wi is joined to vi and vi+1

(

modn

)

. Let e e e1, 2, ,3 ,en be the

consecutive edges of the cycle Cn. e e1′ ′, 2,,en′ are the corresponding edges joining

the apex vertex v0 to the vertices v v1, 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 wi to vi+1

(

modn

)

. Let G

be the graph obtained from SFn by duplication of each of the vertices 0, ,1 2, , n, 1, 2, , n

v v vv w ww by the vertices v v v′ ′ ′0, ,1 2,,v w wn′ ′ ′, 1, 2,,wn′ respectively.

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Let 1, 2, , 2

a a a

n

e ee be the edges joining the vertex wi′ to the adjacent vertex of wi for 1, 2, ,

i=  n. e e1b, 2b,,e5bn 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 ee are the edges joining the vertex v0′ to the

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

( )

=12n and V G

( )

=4n+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 4n+2 vertices. Therefore, vf

( )

0 −vf

( )

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 v0 is the apex vertex, v v1, 2,,vn

be the consecutive rim vertices of Wn and w w1, 2,,wn be the additional vertices

where wi is joined to vi and vi+1

(

modn

)

. Let e e e1, 2, ,3 ,en be the consecutive

rim edges of wn. e e1′ ′, 2,,en′ are the corresponding edges joining apex vertex v0 to

the vertices v v1, 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

(

modn

)

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 uil to vi and r

i

e be the edges joining uir to vi+1

(

modn

)

. Let for each 1≤ ≤i n, l i

f be the edges

joining wi to l i

u and fir be the edges joining wi to r i

u . Thus, E G

( )

=6n 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 1

1 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 2

2 2

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

g n n n

v u u u + u u uw w w

 

 

=  

 

    . So vg

( )

0 =vg

( )

1 + =1 2n+1 and

( )

0

( )

1 3

g g

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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 2

1 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 2

2 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 2n+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 an

edge product cordial labeling on G. So the graph G is an edge product cordial graph. Illustration 4. Graph G obtained from SF8 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 v0 is an edge product cordial graph.

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

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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 u1, 2,,un to the apex

vertex v0. Let v v1, 2,,vn be the rim vertices and w w1, 2,,wn be the pendant

vertices to v v1, 2,,vn respectively.

Let e e e1, 2, ,3 ,en be the consecutive rim edges of Wn. e e1′ ′, 2,,en′ are the cor-

responding edges joining the apex vertex v0 to the vertices v v1, 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

( )

=5n and V G

( )

=3n+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 2

2 2

0 , , , , , 1, 2, ,

f n n n n

v =v v v v u + u + u 

 

    and

( )

1 2 1 2

2

1 , , , , , , , .

f n n

v = w w w u u u 

 

    So

( )

( )

3

0 1 1 1

2

f f

n

v =v + = + and

( )

( )

5

0 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 1

1 2 2 2

0 , , , , , , , ,

f n n n n

v v v v v u ++ u ++ u

 

 

=  

 

    and

( )

1 2 1 2 1

2

1 , , , , , , , .

f n n

v w w w u u u +

 

 

=  

 

    So

( )

( )

3 1

0 1

2

f f

n

v =v = + and

( )

1 5

2

f

n e =   

 ,

( )

5

0 .

2 f

n e =   

 

From both the cases vf

( )

0 −vf

( )

1 ≤1 and ef

( )

0 −ef

( )

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 Fl5 by adding 5 pendent vertices 1, 2, 3, 4, 5

u u u u u to the apex vertex v0 and its edge product cordial labeling is shown in

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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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Figure

Figure 1. SF6 and its edge product cordial labeling

Figure 1.

SF6 and its edge product cordial labeling p.4
Figure 2. G and its edge product cordial labeling.

Figure 2.

G and its edge product cordial labeling. p.5
Figure 3. G and its edge product cordial labeling.

Figure 3.

G and its edge product cordial labeling. p.7
Figure 4. G and its edge product cordial labeling.

Figure 4.

G and its edge product cordial labeling. p.9
Figure 5. 277

Figure 5.

277 p.10
Figure 5. G and its edge product cordial labeling.

Figure 5.

G and its edge product cordial labeling. p.11

References