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### 86

**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: **

**Article history:**

** Abstract **

**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

### }

definedas,

### (

### )

### (

### )

### (mod

### 3

### )

1

###

###

###

###

###

###

###

##

*m*

*i*
*i*

*j*

*f*

*e*

*v*

*f*

, where *ei*

*E*(

*G*)and

*ei*is incident to

*vj*, for

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

1 | ) ( ) (

|*v _{f}*

*i*

*v*

_{f}*j* and |

*e*(

_{f}*i*)

*e*(

_{f}*j*)|1 for

*i*

###

*j*

and *i*,

*j*{0,1,2} , where

)
(*x*

*v _{f}* and

*e*(

_{f}*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: **

**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 **

**Author Correspondence**

**S. Pethanachi Selvam***
Dept. of Mathematics,

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### 87

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

*, cycle*

_{n}*C*

*, star*

_{n}*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 *e*

_{i}*E*(

*G*) and

*e*is incident to

_{i}*vj*, for each

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

0

*n*

*j* Then the map *f* is called Edge sum cordial labeling if|*v _{f}*(

*i*)

*v*(

_{f}*j*)|1

and|*e _{f}*(

*i*)

*e*(

_{f}*j*)|1for

*i*

###

*j*

and*i*,

*j*{0,1}, where

*v*(

_{f}*x*)and

*e*(

_{f}*x*)denote the number of

vertices and edges labeled with*x*,*x*{0,1}.

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

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### 88

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 *ei*

*E*(

*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*

and*i*,

*j*{0,1,2}, where

*vf*(

*x*)and

*ef*(

*x*)denote the number of

vertices and edges labeled with*x*,*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*

*n*and special graphs like Gear graph

*G*

*n*,

double star*K*1,*n*,*n*and Friendship graph*Fn*are at most Edge 3-sum cordial graphs.

0 1 2

0 2

0 1

1

1

0

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### 89

**Theorem 3.1**

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

**Proof **

Let *G**Cn*be a cycle.

Let*V*(*C _{n}*)

###

*v*/0

_{i}*i*

*n*1

###

,*E*

### (

*C*

_{n}### )

###

###

*e*

_{i}### /

### 0

###

*i*

###

*n*

###

### 1

###

. We know that any cycle Cn is an edgesum 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*3

*k*and

*n*

###

### 3

*k*

###

### 1

### ,

where*k*1

Assign the edge labeling as follows.

For each

*i*

###

### 0

### ,

### 1

### ,

### 2

### ,...,

*k*

###

### 1

, we have*f*

### (

*e*

_{3}

_{i}### )

###

### 0

### ,

*f*(

*e*

_{3}

_{i}_{}

_{1})1and

*f*(

*e*

_{3}

_{i}_{}

_{2})2 .

**Case (ii): **

Let

*n*

###

### 2

### (mod

### 3

### )

, where n is odd.Then*n*3*k*2, where k is odd

For each

*i*

###

### 0

### ,

### 6

### ,

### 12

### ,...,

### 3

### (

*k*

###

### 1

### )

, assign edge labeling as follows:,
0
)
(*e _{i}*

*f*

*f*

### (

*e*

_{i}_{}

_{1}

### )

###

### 0

### ,

*f*(

*e*

_{i}_{}

_{2})1,

*f*(

*e*

_{i}_{}

_{3})1,

*f*(

*e*

_{i}_{}

_{4})2and

*f*(

*e*

_{i}_{}

_{5})2 .

**Case (iii): **

Let

*n*

###

### 2

### (mod

### 3

### )

, where n is even. Then*n*

###

### 3

*k*

###

### 2

### ,

where k is evenFix *f*(*e _{n}*

_{}

_{2})2,

*f*(

*e*

_{n}_{}

_{1})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)2and

*f*(

*ei*5)2 .

From the above cases, we have

& 1 | ) ( ) (

|*vf* *i* *vf* *j* |*ef*(*i*)*ef*(*j*)|1,

###

*i*

###

*j*

and *i*

### ,

*j*

###

### {

### 0

### ,

### 1

### ,

### 2

### }

and hence the theorem.

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### 90

**Theorem 3.2**

Any path *P _{n}*,(

*n*3)is at most edge 3-sum cordial graph.

**Proof: **

Let

*G*

###

*P*

*n*be a graph with

###

/0 1###

)

(*P* *v* *i**n*
*V* *n* *i*

and*E*(*Pn*)

###

*ei*/0

*i*

*n*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)and*n*1(mod3)

ie,

*n*

###

### 3

*k*

and*n*

###

### 3

*k*

###

### 1

where*k*

###

### 1

### .

For each*i*0,1,2,...,(

*k*1),assign f(e

_{3}

_{i})1, f(e

_{3}

_{i}

_{}

_{1})2,

.
0
)
(*e*_{3}_{i}_{}_{2}
*f*

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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### 91

**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 each*i*0,1,2,...,(*k*1),assign the edge labeling as follows.

### 1

### )

### (

*e*

3*i*

###

*f*

_{,}

*f*

### (

*e*

3*i*1

### )

###

### 0

,*f*(

*e*3

*i*2)2.

Case (iii):

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

Fix

*f*

### (

*e*

_{n}_{}

_{2}

### )

###

### 2

### .

Assign the edge labeling as follows.2 ) e ( f , 1 ) e ( f , 0 ) e (

f _{3}_{i} _{3}_{i}_{}_{1} _{3}_{i}_{}_{2} ,for each *i*0,1,2,...,(*k*1),

From all the above cases, we have

1 | ) ( ) (

|*e _{f}*

*i*

*e*

_{f}*j* for

*i*

###

*j*

and*i*

### ,

*j*

###

###

### 0

### ,

### 1

### ,

### 2

.**Illustration: **

### :

3

*P*

& 1 | ) ( ) (

|*v _{f}*

*i*

*v*

_{f}*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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### 92

**Theorem 3.3**

Any Gear graph

*G*

*n*, where

*n*

###

### 3

is at most edge 3-sum cordial graph.**Proof: **

Let*G**Gn*be a gear graph having *p*2*n*1vertices and *q*3*n*edges.

Let

*e*

_{0}

### ,

*e*

_{1}

### ,...,

*e*

_{n}_{}

_{1}be the inner edges and e

_{n}v

_{1},e

_{n}

_{}

_{1}v

_{2},...,e

_{3}

_{n}

_{}

_{1}v

_{2}

_{n}be 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): **

Let*n*1(mod3).

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

1 ) ( , 0 ) ( , 2 )

(*ei* *f* *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*(*e _{i}*)1,

*f*(

*e*

_{i}_{}

_{1})2,

*f*(

*e*

_{i}_{}

_{2})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*(*e _{i}*)0,

*f*(

*e*

_{i}_{}

_{1})2,

*f*(

*e*

_{i}_{}

_{2})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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### 93

labeling.**Illusration: **

**Theorem 3.4 **

Any double star *K*1,*n*,*n*is at most edge 3-sum cordial graph.

**Proof: **

Let *G**K*_{1}_{,}_{n}_{,}* _{n}*,(

*n*2)be a graph. Let

*e*

_{0}

### ,

*e*

_{1}

### ,...,

*e*

_{n}_{}

_{1}be the first layer edges in which each

edge having degree n.*en* *v*1,*en*1 *v*2,...,*e*2*n*1 *vn*be the second layer edges of the double star

*n*
*n*

*K*1, , , where

*e*

*n*1is not adjacent to

*e*

*n*. Then|

*V*(

*G*)|2

*n*1&|

*E*(

*G*)|2

*n*.

*f*:

*E*(

*G*){0,1}

induces vertex labeling for the double star graph which is cordial. Clearly, K_{1}_{,}_{n}_{,}_{n} is an edge

sum cordial graph.

Define

*f*

### :

*E*

### (

*G*

### )

###

###

### 0

### ,

### 1

### ,

### 2

. We have two cases.**Case (i): **

Letn0(mod3). Then

*k*

*n*3 where

*k*

###

### 1

. Assign labeling from the first layer*n*

edges. For each*i*0,3,6,...,3(2

*k*1),

we have*f*(*e _{i}*)0,

*f*(

*e*

_{i}_{}

_{1})1,

*f*(

*e*

_{i}_{}

_{2})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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### 94

**Case (ii):**

Let *n*1(mod3)._{ Then } *n*3*k*1,_{ where}

*k*

###

### 1

Let us assign labeling to the first

*n*

edges as follows.
0 ) ( , 2 ) ( , 1 )

(*ei* *f* *ei*1 *f* *ei*2

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

second layer

*n*

edges as in case (i).
**Case (iii): **

Let *n*2(mod3)i.e., n3k2,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* *i* *ef* *j* for

*i*

###

*j*

and*i*

### ,

*j*

###

###

### 0

### ,

### 1

### ,

### 2

and hence*K*1,

*n*,

*n*is at most edge 3-sum cordial

graph.

**Illustration: **

**Theorem 3.5 **

The friendship graph Fl_{n}(n3)is at most edge 3-sum cordial graph.

**Proof: **

Let GFlnbe 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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### 95

Let*e*

0### ,

*e*

1### ,...,

*e*

2*n*1be the inner edges and

*e*2

*n*

*v*1,

*e*2

*n*1

*v*2,

*e*2

*n*2

*v*3,....,

*e*3

*n*1

*vn*

be the outer edges. Then|*V*(*G*)|2*n*1&*E*(*G*)3*n*. *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
)
(*e _{i}*

*f* *f*(*e _{i}*

_{}

_{1})2,

*f*

### (

*e*

_{i}_{}

_{2}

### )

###

### 1

### ,

_{ for each }

*i*0,3,6,9,...,(3

*n*1)

_{. }

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

**Case (ii): **

Whenn2(mod3),start the edge labeling from the inner edges by

2 ) ( , 0 ) ( , 1 )

(*e _{i}*

*f*

*e*

_{i}_{}

_{1}

*f*

*e*

_{i}_{}

_{2}

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

From the above two cases, we have |*v _{f}*(

*i*)

*v*(

_{f}*j*)|1,|

*e*(

_{f}*i*)

*e*(

_{f}*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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### 96

**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 **