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Email: [email protected]. 2 ... Email: [email protected]. Abstract ... Example: The following is a simple example of a difference cordial graph.
Global Journal of Mathematical Sciences: Theory and Practical. ISSN 0974-3200 Volume 5, Number 3 (2013), pp. 185-196 © International Research Publication House http://www.irphouse.com

Difference Cordial Labeling of Graphs R. Ponraj1, S. Sathish Narayanan2 and R. Kala3 1

Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India. Email: [email protected] 2 Department of Mathematics, Thiruvalluvar College, Papanasam-627425, India. Email: [email protected] 3 Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India. Email: [email protected]

Abstract In this paper, we introduce a new notion called difference cordial labeling. Let G be a ( , ) graph. Let : ( ) → {1, 2, … , } be a function. For each edge , assign the label | ( ) − ( ) |. is called a difference cordial labeling if (0) − (1) ≤ 1 where is a one to one map and (1) and (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. Keywords: Path, Cycle, Complete graph, Complete bipartite graph, Star, Helm.

Introduction We consider finite, undirected and simple graphs only. Let = ( , ) be ( , ) graph. The cardinality of the vertex set and edge set respectively is called the order and size of a graph. Cahit introduced the concept of cordial labeling in the year 1987 in [1]. Let be a function from ( ) to {0, 1} and for each edge , assign the label | ( ) − ( ) |. is called a cordial labeling if the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. -Product cordial labeling [4] has been introduced by R. Ponraj, M. Sivakumar and M. Sundaram. Motivated by these works we define here a new notion called difference

186

R. Ponraj, S. Sathish Narayanan and R. Kala

cordial labeling of a graph. Terms and definitions not defined here are used in the sense of Harary [3].

Difference Cordial Labeling Definition 2.1: Let G be a ( , ) graph. Let be a map from ( ) to {1, 2, … , }. For each edge , assign the label | ( ) − ( ) |. is called difference cordial labeling if is 1 − 1 (0) − (1) ≤ 1 where and (1) and (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. Example: The following is a simple example of a difference cordial graph

1 2

4

3 Figure (i)

Theorem 2.2: Every graph is a subgraph of a connected difference cordial graph. Proof: Let G be a given ( , ) graph. Let

=

(

)

− 2 + 2. Consider the

complete graph and the cycle . Let = , ,…, and be the cycle ∗ … . We construct the super graph of G as follows: Let ( ∗ ) = ∗ ( ) ∪ ( ) and ( ) = ( ) ∪ ( ) ∪ . Clearly G is a sub graph of ∗ . Assign the label to (1 ≤ ≤ ) and + to (1 ≤ ≤ ). Therefore, (0) = + and (1) = + − 1. Hence ∗ is a difference cordial graph. ∎ Theorem 2.3: If G is a ( , ) difference cordial graph, then

≤ 2 − 1.

Proof: Let be a difference cordial labeling of G. Obviously, implies (0) ≥ − + 1 →. (1) Case (i) : (0) =

(1) + 1.

(1) ≤

− 1. This

Difference Cordial Labeling of Graphs From (1) , ≤ (0) + − 1 = Therefore, ≤ 2 − 1 → (2) .

187 (1) + 1 +

Case (ii) : (1) = (0) + 1. From (1) , ≤ (0) + − 1 ≤ Therefore, ≤ 2 − 2 → (3) .

(0) +

− 1 ≤ 2 − 2.

Case (ii) : (1) = (0) + 1. From (1) , ≤ (0) + − 1 = (1) − 1 + Therefore, ≤ 2 − 3 → (4) . From (2) , (3) and (4) , ≤ 2 − 1. Theorem 2.4: If

is a -regular graph with

− 1 ≤ 2 − 1.

− 1 ≤ 2 − 3. ∎

≥ 4 then

is not difference cordial.

Proof: Let be a ( , ) graph. Suppose is difference cordial. Then by theorem 2.3, ≤ 2 − 1. This implies ≤ − 1. Hence ≤ − 1. This is impossible. ∎ Theorem 2.5: Any Path is a difference cordial graph. …

Proof: Let be the path cordial labeling of , ≤ 8.

. The following table (i) gives the difference

Table (i) e 1 2 3 4 5 6 7 8

Assume

1 1 1 1 1 1 1 1

2 3 2 2 2 2 2

2 4 4 3 3 3

3 3 5 5 4

5 4 7 6

6 6 8

4 7

5

> 8. Define a map : ( ) → {1, 2, … , } as follows:

Case (i) : ≡ 0 ( Define

4) . +2 ( )= 1≤ ≤ 2 +2 = +2 1≤ ≤ −1 2 4

188

R. Ponraj, S. Sathish Narayanan and R. Kala =

Case (ii) : Define

≡1(

+2 +2 −11≤ ≤ 2 4

4) . +1 ( )= 1≤ ≤ 2 +1 −1 = +2 1 ≤ ≤ 2 4 +1 −1 = +2 −11≤ ≤ 2 4

Case (iii) : Define

≡2(

4) . +2 ( )= 1≤ ≤ 2 +2 −2 = +2 1≤ ≤ 2 4 +2 −2 = +2 −11≤ ≤ 2 4

Case (iv) : Define

≡3(

4) . +1 ( )= 1≤ ≤ 2 +1 +1 = +2 1≤ ≤ −1 2 4 +1 +1 = +2 −11≤ ≤ 2 4

The following table (ii) proves that f is a difference cordial labeling. Table (ii) Nature of n ≡0( 2) ≡1(

2)

(0) −2 2 −1 2

(1) 2 −1 2 ∎

Corollary 2.6: Any Cycle is a difference cordial graph.

Difference Cordial Labeling of Graphs Proof: The function : … .

in theorem 2.5 is also a difference cordial labeling of the cycle ∎

Theorem 2.7: The Star Proof: Let , shows that the star

189

,

≤ 5.

is difference cordial iff

= { , : 1 ≤ ≤ }, ={ , , , ≤ 5 is difference cordial.

:1 ≤ ≤

}. Table (iii)

Table (iii) n 1 2 3 4 5

u 1 1 1 2 2

2 2 2 1 1

3 3 3 3

4 4 4

5 5

6

Assume > 5. Suppose is a difference cordial labeling of , , > 5. Without loss of generality assume that ( ) = . To get the edge label 1, the only possibility is that ( ) = − 1, = + 1 for some , . This implies (1) ≤ 2. (0) − (1) ≥ − 2 − 2 > 1, a contradiction. ∎ Theorem 2.8:

is difference cordial iff

≤ 4. (

)

Proof: Suppose is difference cordial. Then ≤ 2 − 1. This implies ≤ 4. , are difference cordial by theorem 2.5. Using corollary 2.6, is difference cordial. A difference cordial labeling of is given in figure (ii) .

1

4

2

3 Figure (ii) ∎

Now we look into the complete bipartite graph

,

.

190

R. Ponraj, S. Sathish Narayanan and R. Kala ≥ 4 and

Theorem 2.9: If Proof: Suppose This implies and ≥ 4. ∎ Theorem 2.10:

,

,

≥ 4, then

is not difference cordial.

,

is difference cordial. By theorem 2.3, ≤ 2( + ) − 1. − 2 − 2 + 1 ≤ 0, a contradiction to ≥4

≤ 4.

is difference cordial iff

Proof: Let = ∪ where = { , } and = { : 1 ≤ ≤ }. , , , , are difference cordial by theorem 2.5 and corollary 2.6 respectively. A difference cordial labeling of , and , are given in figure (iii) .

1

4

2

2

3

1

5

4

5

3

6

Figure (iii) Now, we assume ≥ 5. Suppose is a difference cordial labeling. Let ( ) = , ( ) = . Then , has at most 4 edges with label 1. The maximum value is attained if the vertices in the set receive the labels − 1, + 1, − 1, + 1. (1) ≤ 4, (0) ≥ 2 − 4. Hence (0) − (1) ≥ 2 − 8 > 2, a Therefore contradiction. ∎ Theorem 2.11:

,

is difference cordial iff

≤ 4.

Proof: , , , are difference cordial graphs by theorem 2.7 and 2.10. A difference cordial labeling of , , , are given in figure (iv)

1

2

3

4

6

1

Figure (iv)

6

4

2

5

3

5

7

Difference Cordial Labeling of Graphs

191

For any injective map f on (1) ≤ 6. Therefore, , is not difference , , cordial. Assume ≥ 6. Suppose , is difference cordial, then by theorem 2.3, 3 ≤ 2( + 3) − 1, a contradiction. ∎ Now we investigate the difference cordial labeling behavior of bistar +

Theorem 2.12: If

Proof: Let , , , ∶1≤ ≤

≥ 9 then

,

.

,

is not difference cordial.

= , , , ∶ 1 ≤ ≤ ,1 ≤ ≤ and , 1 ≤ ≤ . Assume ( ) = and ( ) = .

,

=

Case (i) : ≠ − 1 and ≠ + 1. To get the edge label 1, and must receive the labels − 1, + 1 respectively for some , and , must receive the labels − 1, + 1 respectively for some , . Hence (1) ≤ 4. Case (ii) : = − 1 or + 1. Obviously, in this case (1) ≤ 3. Thus by case (i) , (ii) , (1) ≤ 4. Therefore, (0) ≥ − 4 ≥ + − 3. (0) − (1) ≥ + − 3 − 4 ≥ 2, Then a contradiction. ∎ Theorem 2.13: Proof: Let ≤ }.

,

,

is difference cordial iff ={ , ,

,

≤ 5.

∶ 1 ≤ ≤ } and

,

={

,

,

∶1≤

Case (i) : ≤ 5. , is difference cordial by theorem 2.4. A difference cordial labeling of , is in figure (v)

2

3

5

4

1 Figure (v)

For 3 ≤ ≤ 5, define ( ) = 2, ( ) = 4, ( ) = 1, ( ) = 3, ( ) = 3 + , 2 ≤ ≤ . Clearly this f is a difference cordial labeling.

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R. Ponraj, S. Sathish Narayanan and R. Kala

Case (ii) : > 5. When ≥ 8, the result follows from theorem 2.12. When = 6 7, (1) ≤ 3. Therefore, (0) ≥ 5 6 according as = 6 7. This is a contradiction. ∎ Theorem 2.14:

,

Proof: ,

is difference cordial iff

Let ={

,

≤ 6.

={ , , ∶ 1 ≤ ≤ }. ,

,

,

,

,

∶1≤ ≤ }

and

Case (i) : = 2, 3. When = 2, define ( ) = 4, ( ) = 2, ( ) = 6, ( ) = 5, ( ) = 1, ( ) = 3. When = 3, define ( ) = 5, ( ) = 2, ( ) = 6, ( ) = 7, ( ) = 1, ( ) = 3, ( ) = 4. Clearly, f is a difference cordial labeling. Case (ii) : = 4, 5, 6. ( ) = 2, ( ) = 1, ( ) = 1 + , 2 ≤ ≤ , ( ) = + 2, ( ) = + Define (0) = − 1 and (1) = 4. Therefore, f is a 3, ( ) = + 4. In this case, difference cordial labeling. Case (iii) : ≥ 7. Proof follows from theorem 2.12. Theorem 2.15:

,

is difference cordial iff

∎ ≤ 5.

Proof: Let = , , , ∶ 1 ≤ ≤ 3, 1 ≤ ≤ and = , , , , ∶ 1 ≤ ≤ 3, 1 ≤ ≤ . , , , are difference cordial by theorems 2.13 and 2.14 respectively. For 3 ≤ ≤ 5, define ( ) = 2, ( ) = 1, ( ) = 1 + , 2 ≤ ≤ , ( ) = + 2, ( ) = + 3, ( ) = + 4, ( ) = + 5. In this case, (0) = and (1) = 4. Therefore, f is a difference cordial labeling. For ≥ 6, the result follows from theorem 2.12. ∎ Remark 2.16:

,

is difference cordial.

Theorem 2.17: The wheel

is difference cordial.

Proof: Let = + where is the cycle … and ( ) = { }. Define a map : ( ) → {1, 2, … + 1} by ( ) = 1, ( ) = + 1, 1 ≤ ≤ . Then (0) = , (1) = . ∎ Corollary 2.18: The fan

is difference cordial for all n.

Proof: Let = + where is the path … and ( ) = { }. The function f given in theorem 2.17 is also a difference cordial labeling since (0) =

Difference Cordial Labeling of Graphs − 1,

193

(1) = .



The gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle . Theorem 2.19: The gear graph is difference cordial. Proof: Let the vertex set and edge set of the wheel be defined as in theorem 2.17. Let ( ) = ( ) ∪ { ∶ 1 ≤ ≤ } and ( ) = ( ) ∪ , ∶1≤ ≤ , 1 ≤ ≤ } − ( ). Define : ( ) → {1, 2 … 2 + 1} as follows: Case (i) : n is even. ( ) =2 −11≤ ≤ 1≤ ≤ (

≡0( )=

1≤ ≤

≡2( 4)

3 +3+ 1 ≤ ≤ 2 ≡0( 4)

( ) =2 1≤ ≤ 3 −4 1≤ ≤ ≡0( 4 ( ) = + 1≤ ≤ 1≤ ≤

≡0(

−2 4

(

(

4) ≡2(

1≤ ≤

4)

≡1(

4)

4) ≡1(

4)

+3+ 1≤ ≤

≡1(

4)

≡1(

≡3( )=

4)

+3+ 1≤ ≤

( )=2 1≤ ≤ 1≤ ≤

4)

4)

≡3( )=

≡2(

≡2(

Case (ii) : n is odd. ( )=2 −11≤ ≤ 1≤ ≤

4)

4)

+ −11≤ ≤ ≡0(

Finally we define cordial labeling.

4) ≡2(

4)

4) ( ) = 2 + 1. Table (iv) establishes that

is a difference

194

R. Ponraj, S. Sathish Narayanan and R. Kala Table (iv) (0) (1) 3 3 2 2 3 −1 3 +1 2 2

Values of n ≡0( 2) ≡1(

2)



We now look into helms and webs. The helm is obtained from a wheel by attaching a pendent edge at each vertex of the cycle . Koh et al. [2] define a web graph. A web graph is obtained by joining the pendent points of the helm to form a cycle and then adding a single pendent edge to each vertex of this outer cycle. Yang [2] has extended the notion of a web by iterating the process of adding pendent vertices and joining them to form a cycle and then adding pendent point to the new cycle. ( , ) is the generalized web with t cycles . Note that (1, ) is the helm and (2, ) is the web. Theorem 2.20: All webs are difference cordial. Proof: Let

()



be the cycle

( , ) =

. Let

()

∪{ }∪{

()



∶1≤ ≤ }

And ( , ) = ∪

∶1≤ ≤

(

if if

is even if

is even

is odd

)=4 1≤ ≤

1≤ ≤

if

is odd

)=4 −31≤ ≤

1≤ ≤ (

if

+ 1} as follows:

is odd

)=4 −11≤ ≤

1≤ ≤ (

if

∶1≤ ≤

∶ 1 ≤ ≤ ,1 ≤ ≤ − 1 .

( , ) → {1, 2 … + Define a map : ( ) = 4 − 2 1 ≤ ≤ if is even 1≤ ≤



if

is even

is odd

= ( + 1) + + 1 ≤ ≤ − 1, 1 ≤ ≤

− .

Difference Cordial Labeling of Graphs

195

= ( + 1) + − 1 ≤ ≤ − 1, 0 ≤ ≤ − 1. ( ) = + + 1. From the following table (v) , f yields a difference cordial labeling of the webs ( , ) Table (v) Nature of n ≡0( 2) ≡1(

2)

(0) (2 + 1) 2 (2 + 1) − 1 2

(1) (2 + 1) 2 (2 + 1) + 1 2 ∎

(3, 5) with a difference cordial labeling is given in figure (vi)

A web graph

Figure (vi)

Corollary 2.21: The helm Proof: Since



is difference cordial for all n.

(1, ) , the proof follows from theorem 2.20.



References [1] I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars combinatoria 23 (1987) , 201-207. [2] J. A. Gallian, A Dynamic survey of graph labeling, the electronic journal of combinatorics 18 (2013) # Ds6.

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R. Ponraj, S. Sathish Narayanan and R. Kala

[3] F. Harary, Graph theory, Addision wesley, New Delhi (1969) . [4] R. Ponraj, M. Sivakumar and M. Sundaram, −Product cordial labeling of graphs, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 15, 733-742.