AT MOST EDGE 3 – SUM CORDIAL LABELING FOR SOME GRAPHS THE STANDARD

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AT MOST EDGE 3 - SUM CORDIAL LABELING FOR SOME GRAPHS

THE STANDARD

S. Pethanachi Selvam* S. Padmashini**

1.

Introduction

Throughout this paper we consider finite, simple and undirected graphs. Let

G

(

V

,

E

)

be a graph, where

V

(

G

)

and

E

(

G

)

respectively denote the vertex set and edge set of G. Also, for

Article history:

Abstract

Received July 9th, 2016; Approved July 15th , 2016; Available online: Aug 5th, 2016.

Let

G

(

V

,

E

)

be a graph of order n and size m. An edge labeling function

}

2

,

1

,

0

{

)

(

:

E

G

f

induces a vertex labeling function

f

:

V

(

G

)

{

0

,

1

,

2

}

defined

as,

(

)

(

)

(mod

3

)

1

m

i i

j

f

e

v

f

, where eiE(G)and eiis incident tovj, for

each j0,1,2,...,(n1).Then the map f is called Edge 3-sum cordial labeling if

1 | ) ( ) (

|vf ivf j  and |ef(i)ef(j)|1 for

i

j

and i,j{0,1,2} , where

) (x

vf and ef(x) denote the number of vertices and edges labeled with

} 2 , 1 , 0 { ,x

x .An edge 3-sum cordial graph which admits an edge sum cordial labeling is called at most edge 3-sum cordial labeling. A graph having at most edge 3-sum cordial labeling is called at most edge 3-sum cordial graph. In this paper, we prove that some standard graphs and special graphs like Gear graph, Double star and Friendship graph are at most edge 3-sum cordial graphs.

Keywords:

Edge 3-sum,

At most edge 3-sum cordial graph.

2395-7492© Copyright 2016 The Author. Published by International Journal of Engineering and Applied Science. This is an open access article under the

All rights reserved.

Author Correspondence

S. Pethanachi Selvam* Dept. of Mathematics,

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a graph G, p and q denote the number of vertices and edges respectively. If the vertices or edges

or both of the graphs are assigned values subject to certain conditions, it is known as graph

labeling. A detailed study on variety of applications on graph labeling is carried out in Bloom and

Golomb[1]. Cordial graphs were first introduced by Cahit[2] as a weaker version of both graceful

and harmonious graphs.

The product cordial labeling and k-product cordial labeling of graphs was introduced by

Ponraj.R, Sivakumar.M and Sundaram.M [4,6], in which they proved that some standard and

special graphs admit k-product cordial labeling. The concept of sum cordial labeling was

introduced by J.Shiama [5] in which she proved that some graphs like path

P

n, cycle

C

n, star

n

K

1, ,etc., are sum cordial graphs. Motivated by these papers, we define edge 3-sum and at most

edge 3-sum cordial labeling of G.

2.

Edge Sum and At most Edge 3-sum Cordial Labeling

Let

G

(

V

,

E

)

be a graph of order n and size m. An edge labeling function

}

1

,

0

{

)

(

:

E

G

f

induces a vertex labeling function

f

:

V

(

G

)

{

0

,

1

}

defined as,

)

2

(mod

)

(

)

(

1

m

i i

j

f

e

v

f

, where eiE(G) and ei is incident to vj , for each

). 1 ( ,..., 2 , 1 ,

0 

n

j Then the map f is called Edge sum cordial labeling if|vf(i)vf(j)|1

and|ef(i)ef(j)|1for

i

j

andi,j{0,1}, wherevf(x)andef(x)denote the number of

vertices and edges labeled withx,x{0,1}.

Dept. of Mathematics ,The Standard Fireworks Rajaratnam College for Women, Sivakasi (Tamilnadu)*

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Let

G

(

V

,

E

)

be a graph of order n and size m. An edge labeling function

}

2

,

1

,

0

{

)

(

:

E

G

f

induces a vertex labeling function

f

:

V

(

G

)

{

0

,

1

,

2

}

defined as,

)

3

(mod

)

(

)

(

1

m

i i

j

f

e

v

f

, where eiE(G) and ei is incident to vj , for each

). 1 ( ,..., 2 , 1 ,

0 

n

j Then the map f is called Edge 3-sum cordial labeling if|vf(i)vf(j)|1

and|ef(i)ef(j)|1for

i

j

andi,j{0,1,2}, wherevf(x)andef(x)denote the number of

vertices and edges labeled withx,x{0,1,2}.

An edge 3-sum cordial graph which admits an edge sum cordial labeling is called at most

edge 3-sum cordial labeling. A graph having at most edge 3-sum cordial labeling is called at most

edge 3-sum cordial graph.

Note:

Every at most edge 3-sum cordial graph is edge 3-sum cordial graph. But the converse need

not be true. For example, consider the 5-pan graph.

It is an edge 3-sum cordial graph. But it is not at most edge 3-sum cordial graph, as it does not

admit edge sum cordial labeling.

3.

Main Results

We prove that some standard graphs

P

n,

C

nand special graphs like Gear graph

G

n,

double starK1,n,nand Friendship graphFnare at most Edge 3-sum cordial graphs.

0 1 2

0 2

0 1

1

1

0

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Theorem 3.1

Any cycle Cn,(n3)is at most Edge 3-sum cordial graph.

Proof

Let GCnbe a cycle.

LetV(Cn)

vi /0in1

,

E

(

C

n

)

e

i

/

0

i

n

1

. We know that any cycle Cn is an edge

sum cordial graph. Define f :E(G)

0,1,2

. Consider the following three cases.

Case (i):

Let

n

0

(mod

3

)

and

n

1

(mod

3

)

. Then n3kand

n

3

k

1

,

wherek1

Assign the edge labeling as follows.

For each

i

0

,

1

,

2

,...,

k

1

, we have

f

(

e

3i

)

0

,

f(e3i1)1and f(e3i2)2 .

Case (ii):

Let

n

2

(mod

3

)

, where n is odd.

Thenn3k2, where k is odd

For each

i

0

,

6

,

12

,...,

3

(

k

1

)

, assign edge labeling as follows:

, 0 ) (ei

f

f

(

e

i1

)

0

,

f(ei2)1, f(ei3)1, f(ei4)2andf(ei5)2 .

Case (iii):

Let

n

2

(mod

3

)

, where n is even. Then

n

3

k

2

,

where k is even

Fix f(en2)2,f(en1)0.

For

i

0

,

6

,

12

,...,

3

(

k

2

)

, assign the edge labeling as follows.

, 0 ) (ei

f

f

(

e

i1

)

0

,

f(ei2)1, f(ei3)1, f(ei4)2andf(ei5)2 .

From the above cases, we have

& 1 | ) ( ) (

|vf ivf j  |ef(i)ef(j)|1,

i

j

and

i

,

j

{

0

,

1

,

2

}

and hence the theorem.

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Theorem 3.2

Any path Pn,(n3)is at most edge 3-sum cordial graph.

Proof:

Let

G

P

nbe a graph with

/0 1

)

(PvinV n i

andE(Pn)

ei /0in1

. We know that any path Pn(n3)is an edge sum cordial graph.

Define

f

:

E

(

G

)

 

0

,

1

,

2

.

We consider the following three cases.

Case (i):

Let n0(mod3)andn1(mod3)

ie,

n

3

k

and

n

3

k

1

where

k

1

.

For eachi0,1,2,...,(k1),assign f(e3i)1, f(e3i1)2,

. 0 ) (e3i2f

1 0

0

0 2 0

2

1 2 1 0 2

0

0 1 2

1 0 2

1 2

0 2

1 0

1

0 2

1 0

2

1

1

0 0

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Case (ii):

Let n2(mod3)where

n

is odd.

ie,

n

3

k

2

where 𝑘 is odd.

Fix f(en2)0and for eachi0,1,2,...,(k1),assign the edge labeling as follows.

1

)

(

e

3i

f

,

f

(

e

3i1

)

0

, f(e3i2)2.

Case (iii):

Let n2(mod3),where n is even. ie, 𝑛 = 3𝑘 + 2, when 𝑘 is even

Fix

f

(

e

n2

)

2

.

Assign the edge labeling as follows.

2 ) e ( f , 1 ) e ( f , 0 ) e (

f 3i3i13i2  ,for each i0,1,2,...,(k1),

From all the above cases, we have

1 | ) ( ) (

|ef ief j  for

i

j

and

i

,

j

 

0

,

1

,

2

.

Illustration:

:

3

P

& 1 | ) ( ) (

|vf ivf j

0 2

1

1 0 2 0

2 1

0 0 1 2 2

1 0

1 2

0 1

0 2

2 2

1

1 0

2 0

1 0

1 2

2

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Theorem 3.3

Any Gear graph

G

n, where

n

3

is at most edge 3-sum cordial graph.

Proof:

LetGGnbe a gear graph having p2n1vertices and q3nedges.

Let

e

0

,

e

1

,...,

e

n1be the inner edges and env1,en1v2,...,e3n1v2nbe the outer edges.

} 1 , 0 { ) ( :E G

f induce vertex labeling function for the gear graph which is cordial. Clearly, Gn is

an edge sum cardial graph. Define

f

:

E

(

G

)

 

0

,

1

,

2

Consider the following cases.

Case (i):

Letn1(mod3).

For each i0,3,6,...,3(n1) , we can assign labeling to the inner edges as

1 ) ( , 0 ) ( , 2 )

(eif ei1  f ei2 

f and continuing the labeling to one of the outer edges which is

adjacent to the last inner edge.

Case (ii):

Let n0(mod3). For each 𝑖 = 0, 3, 6, … . . , 3(𝑛 − 1), we assign labeling as follows.

f(ei)1, f(ei1)2, f(ei2)0 and continue labeling as in case (i)

Case (iii):

When n2(mod3), let us start labeling from the inner edges. For each i0,3,6,...,3(n1),

assign f(ei)0, f(ei1)2, f(ei2)1and continuing the labeling to one of the outer edges

which is adjacent to the last inner edge.From, all the above cases, we have|vf(i)vf(j)|1 and

j )} j ( e ) i ( e

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labeling.

Illusration:

Theorem 3.4

Any double star K1,n,nis at most edge 3-sum cordial graph.

Proof:

Let GK1,n,n,(n2)be a graph. Let

e

0

,

e

1

,...,

e

n1be the first layer edges in which each

edge having degree n.env1,en1 v2,...,e2n1 vnbe the second layer edges of the double star

n n

K1, , , where

e

n1is not adjacent to

e

n. Then|V(G)|2n1&|E(G)|2n. f :E(G){0,1}

induces vertex labeling for the double star graph which is cordial. Clearly, K1,n,n is an edge

sum cordial graph.

Define

f

:

E

(

G

)

 

0

,

1

,

2

. We have two cases.

Case (i):

Letn0(mod3). Then

k

n3 where

k

1

. Assign labeling from the first layer

n

edges. For eachi0,3,6,...,3(2k1),

we havef(ei)0, f(ei1)1, f(ei2)2and continue the labeling to the second layer

n

edges

also.

0 2 2

2

1

1

1 G5 0

2 2 2 0 1 1 1

2 2

0

1

0 1

1 0

0 2

1

0 2 2

0

0

1 G3 1

2 1

2 0

1 0

2

2

0

2 1

0 2

2 0 1 2

1

1

0

1

1 2 2 0

1 0

2 0

0

1

2 G4

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Case (ii):

Let n1(mod3). Then n3k1, where

k

1

Let us assign labeling to the first

n

edges as follows.

0 ) ( , 2 ) ( , 1 )

(eif ei1  f ei2 

f for each i0,3,6,...,(6k) , and continue labeling to the

second layer

n

edges as in case (i).

Case (iii):

Let n2(mod3)i.e., n3k2,where

k

1

.

For each i = 0,3,6,…,(6k), assign labeling as in case (i).

From the above cases, we have|vf(i)vf(j)|1

1 | ) ( ) (

|ef ief j  for

i

j

and

i

,

j

 

0

,

1

,

2

and hence K1,n,nis at most edge 3-sum cordial

graph.

Illustration:

Theorem 3.5

The friendship graph Fln(n3)is at most edge 3-sum cordial graph.

Proof:

Let GFlnbe a friendship graph.

0 1

0

0

1 2

0 2

0

1

1

2

2

1

2 0 1 2

0 2

2

1

0 1

2 0 1

0

2 1

1 2 1

0 1 0

2 1 0 2 1

2 0 1 2 0

2 0 1 2 0

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Let

e

0

,

e

1

,...,

e

2n1be the inner edges and e2nv1,e2n1 v2,e2n2 v3,....,e3n1 vn

be the outer edges. Then|V(G)|2n1&E(G)3n. f :E(G){0,1}induces vertex labeling

for the friendship graph which is cordial. Clearly, Fnis edge sum cordial graph.

Define

f

:

E

(

G

)

 

0

,

1

,

2

.

We prove the following cases.

Case (i):

Let n0(mod3)&n1(mod3)

Start edge labeling from the inner edges by

, 0 ) (ei

f f(ei1)2,

f

(

e

i2

)

1

,

for each i0,3,6,9,...,(3n1).

The outer edges are labeled by the remaining number which is not labeled in each leaf.

Case (ii):

Whenn2(mod3),start the edge labeling from the inner edges by

2 ) ( , 0 ) ( , 1 )

(eif ei1f ei2

f , for i0,3,6,9,...,3(n1).Then continue labeling as in case (i).

From the above two cases, we have |vf(i)vf(j)|1,|ef(i)ef(j)|1 for

i

j

and

 

0

,

1

,

2

,

j

i

. Illustration: 2 2 2 0 0 0

1 2 1 1 1 1 1 0

0 0

2

2

2 0

1

2 2

0 1

1 0

2 0 1 0 2 1 2 0 1 0 2 1 2 0

1 2 0 1 2 0 1 2 0 2 0 1

0 1 0 0 2 2 1 1 0 0 2 2 1

2 1

0 2 1

1

0

F

3

F

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References

[1] Bloom.G.S, Golomb .S.W., 1977, Applications of numbered undirected graph, Proceeding of IEEE, 65(4), pp.562 – 570.

[2] Cahit .I., 1987, Cordial Graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, Vol.23, pp.201 – 207.

[3] Frank Harrary., 2001, Graph Theory, Narosa Publishing House.

[4] Ponraj.R, Sivakumar.M and Sundaram .M., 2012, k-Product cordial labeling of graphs, Int. J. Contemp. Math. Sciences, 7 (15), 733 – 742.

[5] Shiama .J., 2012, Sum cordial labeling for some graphs, IJMA-3(9).

[6] Sundara.m Ponraj .M and Soma Sundaram .S., 2004, Product cordial labeling of graphs, Bull Pure and Applied Sciences (Mathematics and Statistic), Vol.23E,

pp.155 – 163. Vaidya, Dani N.A, 2010, “Some new product cordial graphs”, Journal of App. Comp. Sci. Math., 8 (4), pp. 62-65.

AT MOST EDGE 3 - SUM CORDIAL LABELING FOR SOME GRAPHS The Standard

Figure

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