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
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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
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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
,
.
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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.