PAIR DIFFERENCE CORDIAL LABELING OF GRAPHS

Let G = (V,E) be a (p,q) graph. Define ρ =  p 2 , if p is even p−1 2 , if p is odd and L = {±1,±2,±3, · · · ,±ρ} called the set of labels. Consider a mapping f : V −→ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling | f (u)− f (v)| such that ∣∣∣∆ f1 −∆ f c 1 ∣∣∣ ≤ 1, where ∆ f1 and ∆ f c 1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of path, cycle, star, comb.


INTRODUCTION
In this paper we consider only finite, undirected and simple graphs.The notion of diference cordial labeling of a graph was introduced and studied some properties of difference cordial labeling in [4] .The difference cordial labeling behavior of several graphs like path, cycle, star etc have been investigated in [4].In this paper we introduce the pair difference cordial labeling and investigate pair difference cordial labeling behavior of path,cycle,star,comb and bistar graph.

PRELIMINARIES
Definition 2.1. The ladder L n is the product graph P n XK 2 with 2n vertices and 3n − 2 edges.
Definition 2.2. The graph obtained by joining two disjoint cycles u 1 u 2 , · · · u m u 1 and v 1 v 2 , · · · v n v 1 with an edge u 1 v 1 is called dumbbell graph and it is denoted by Db(m, n). Consider a mapping f : V −→ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling | f (u) − f (v)| such that

PAIR DIFFERENCE CORDIAL LABELING
, where ∆ f 1 and ∆ f c 1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1.A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph.
Case 2. p is odd.
In this case,one vertex label is repeated.This vertex label contributes maximum two edges with Theorem 3.2. The path P n is pair difference cordial for all values of n except n = 3 .
Proof. Let P n be the path u 1 u 2 · · · u n .
Case. 1 n is odd.
There are two cases arises.
Assign the labels 1, 2 to the vertices u 1 , u 2 respectively and assign the labels −1, −2 respectively to the vertices u 3 , u 4 .Next assign the labels 3, 4 respectively to the vertices u 5 , u 6 and assign the labels −3, −4 to the vertices u 7 , u 8 respectively.Proceeding like this untill we reach the vertex u n−1 .Finally assign the label −2 to the vertex u n .Note that the vertices u n−4 , u n−3 get the labels n−3 2 , n−1 2 respectively and the vertices u n−2 , u n−1 receive the labels − n−3 2 , − n−1 2 respectively. This vertex labeling gives the pair difference cordial labeling of path P n ,since ∆ f 1 = ∆ f c 1 = n−1 2 .
Assign the labels 1, 2 respectively to the vertices u 1 , u 2 and assign the label −1, −2 to the vertices u 3 , u 4 respectively.Next assign the labels 3, 4 respectively to the vertices u 5 , u 6 and assign the labels −3, −4 to the vertices u 7 , u 8 respectively.Proceeding like this untill we reached u n−3 .Assign the label − n−3 2 to the vertex u n .Finally assign the labels n−1 2 , − n−1 2 respectively to the vertices u n−2 , u n−1 .Note that the vertices u n−6 , u n−5 received the labels n−5 2 , n−3 2 respectively and the vertices u n−4 , u n−3 get the labels − n−5 2 , − n−3 2 respectively. This vertex labeling gives the pair difference cordial labeling of path P n ,since Subcase. 3 n = 3 .
Suppose f is a pair difference cordial of P 3 , then ∆ f 1 = 0 and ∆ f c 1 = 2. This contradicts P 3 is not pair difference cordial.
Case. 2 n is even.
There are two cases arises.
Assign the labels 1, 2 to the vertices u 1 , u 2 respectively and assign the labels −1, −2 to the vertices u 3 , u 4 respectively.Next assign the labels 3, 4 to the vertices u 5 , u 6 respectively and assign the labels −3, −4 respectively to the vertices u 7 , u 8 .Proceeding like this untill we reach the vertex u n .Note that the vertices u n−3 , u n−2 respectively receive the labels n−2 2 , n 2 and the vertices u n−1 , u n get the labels − n−2 2 , − n 2 respectively. This vertex labeling gives a pair difference cordial labeling of the path P n ,since Assign the labels 1, 2 respectively to the vertices u 1 , u 2 .Now assign the labels −1, −2 to the vertices u 3 , u 4 respectively.Next assign the label 3, 4 respectively to the vertices u 5 , u 6 and assign the label −3, −4 to the vertices u 7 , u 8 respectively.Proceeding like this until we reach the vertex u n−2 .Finally assign the labels n 2 , − n 2 to the vertices u n−1 , u n respectively.Note that the vertices u n−5 , u n−4 get the label n−4 2 , n−2 2 respectively and the vertices u n−3 , u n−2 receive the labels − n−4 2 , − n−2 2 respectively. This vertex labeling gives the pair difference cordial labeling of path P n ,since Remark. P 3 is difference cordial but not pair difference cordial [4]. Proof. Let C n be the cycle u 1 u 2 · · · u n u 1 . The function f in the theorem 3.3 is also a pair difference cordial labeling of the cycle C n .
Theorem 3.3. The star K 1,n is pair difference cordial if and only if 3 ≤ n ≤ 6.
The graph K 1,n has n + 1 vertices and n edges. Table 1 shows that the star K 1,n , 3 ≤ n ≤ 6 is pair difference cordial.
Suppose f is a pair difference cordial labeling of K 1,n .Assume f (u) = l,To get the edge label 1, the only possibly is that the pendant vertices receive the label l − 1 or l + 1.
In this case, we may use one vertex label as twice.This implies Remark. The star K 1,6 is pair difference cordial but not difference cordial [4]. Proof. Case 1. p ≤ 2.
The Table 2 shows that K 3 , K 4 , K 5 is not pair diffrence cordial.
Suppose f is a pair difference cordial.Assume f (u) = l 1 and f (v) = l 2 .To get the edge label 1, the only possibly is that the vertices with degree two receive the label l 1 − 1 or l 1 + 1 and l 2 − 1 or l 2 + 1. Subcase 1.m is even.
In this case we may use one vertex label as twice. This implies Theorem 3.6. Th bistar B 1,n is pair difference cordial if and only if 2 ≤ n ≤ 6. Table 3 shows that the bistar B 1,n , 2 ≤ n ≤ 6 is pair difference cordial.
This implies ∆ f c There are two cases arises. Case 1. m + n ≤ 9.
There are two subcase arises.
Next assign the remaining labels to the remaining vertices in any order.
There are two subcase arises.
When m + n is odd, either m or n is odd.Hence one vertex label is repeated. Therefore Therefore B m,n , m + n ≥ 10 is not pair difference cordial.
Theorem 3.8. The laddar graph P 2 × P n is pair difference cordial for all values of n.
Assign the labels 1, 2 to the vertices v 1 , v 2 respectively.Next assign the labels 3, 5 respectively to the vertices v 3 , v 4 and assign the labels 4, 6 to the vertices v 5 , v 6 respectively.Now assign the labels 7, 9 to the vertices v 7 , v 8 respectively and assign the labels 8, 10 to the vertices v 9 , v 10 respectively.Proceeding like this until we reach v n .Note that in this process the vertex v n get the label n − 1.
As in Subcase 1, assign the labels to the vertices v i , (1 ≤ i ≤ n).Here the vertex v n receive the label n − 1.  Similar to Subcase 1 assign the labels to the vertices v i , (1 ≤ i ≤ n).Note that the vertex v n receive the label n.
The Table 4 given below establish that this vertex labeling f is a pair difference cordial of P n × P 2 .
Nature of n ∆ f c  Theorem 3.9. The dumbbell graph Db(n, n) is pair difference cordial for all values n.
Proof. The vertex set and the edge set of Db(n, n) is given in definition 2.2.
There are four cases arises.
Nature of n ∆ f 1 ∆ f c 1 n ≡ 0 (mod 4) n + 1 n n ≡ 1 (mod 4) n n + 1 n ≡ 2 (mod 4) n n + 1 n ≡ 3 (mod 4) n + 1 n TABLE 5 Assign the labels 1, 2 respectively to the vertices u 1 , u 2 then assign the labels 4, 3 to the vertices u 3 , u 4 .Secondly assign the labels 5, 6 to the vertices u 5 , u 6 then assign the labels 8, 7 to the vertices u 7 , u 8 .Proceeding like this until we reach the vertex u n .Next assign the label 2 to the vertex u n+1 .Now we consider the vertices v i , 1 ≤ i ≤ n. Assign the labels −1, −2 to the vertices  As in case 1, assign the labels to the vertices u i , 1 ≤ i ≤ 5 and v i , 1 ≤ i ≤ 5.Finally assign the label 1 to the vertex u 5 .
As in case 1, assign the labels to the vertices u i , 1 ≤ i ≤ n + 1 and v i , 1 ≤ i ≤ n.  There are four cases arises.
Assign the labels −1, −2 respectively to the vertices v 1 , v 2 and assign the labels −4, −3 to the vertices v 3 , v 4 respectively.Secondly assign the labels −5, −6 to the vertices v 5 , v 6 respectively.Next assign the labels −8, −7 to the vertices v 7 , v 8 respectively.Proceeding like this until we reach the vertex v n .Note that in this the vertex v n receive the label −n + 1.Next cosider the vertices u i , 1 ≤ i ≤ m.
Assign the labels 1, 2 to the vertices u 1 , u 2 respectively and assign the labels 4, 3 respectively to the vertices u 3 , u 4 .Now assign the labels 5, 6 to the vertices u 5 , u 6 respectively and assign the labels 8, 7 respectively to the vertices u 7 , u 8 .Proceeding like this until we reach the vertex u n .Finally consider the remaining m − n vertices.There are four cases arises.
Assign the labels n + 1, n + 2 to the vertices u n+1 , u n+2 respectively and assign the labels −n − 1, −n − 2 respectively to the vertices u n+3 , u n+4 .Secondly assign the labels n + 3, n + 4 to the vertices u n+5 , u n+6 respectively.Next assign the labels −n−3, −n−4 respectively to the vertices u n+7 , u n+8 .Proceeding like this until we reach the vertex u m .  Assign the labels as in subcase 1 to the vertices u i , 1 ≤ i ≤ m − 1.Next assign the label m+n 2 to the vertex u m . Assign the labels as in subcase 1 to the vertices u i , 1 ≤ i ≤ m − 3 and lastly assign the labels − m+n 2 , m+n 2 , 2 respectively to the vertices u m−2 , u m−1 , u m . The Table 6 given below establish that this vertex labeling f is a pair difference cordial of Db(m, n).  Assign the labels as in case 1 to the vertices v i , (1 ≤ i ≤ n).Here note that the vertex v n receive the label −n + 1.
Next consider the remaining m − n vertices.There are four cases arises. As in case 1, assign the labels to the vertices u i , 1 ≤ i ≤ m.
Assign the labels as in subcase 1 to the vertices u i , 1 ≤ i ≤ m − 1.Next assign the label 2 to the vertex u m .