Some New Results on Prime Cordial Labeling

A prime cordial labeling of a graphGwith the vertex setV(G) is a bijection f : V(G) → {1, 2, 3, . . . , |V(G)|} such that each edge uV is assigned the label 1 if gcd(f(u), f(V)) = 1 and 0 if gcd(f(u), f(V)) > 1; then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits a prime cordial labeling is called a prime cordial graph. In this work we give a method to construct larger prime cordial graph using a given prime cordial graph G. In addition to this we have investigated the prime cordial labeling for double fan and degree splitting graphs of path as well as bistar. Moreover we prove that the graph obtained by duplication of an edge (spoke as well as rim) in wheelW n admits prime cordial labeling.


Introduction
We consider a finite, connected, undirected, and simple graph = ( ( ), ( )) with vertices and edges which is also denoted as ( , ). For standard terminology and notations related to graph theory we follow Balakrishnan and Ranganathan [1] while for any concept related to number theory we refer to Burton [2]. In this section we provide brief summary of definitions and other required information for our investigations. Definition 1. The Graph labeling is an assignment of numbers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges), then the labeling is called a vertex labeling (edge labeling).
Many labeling schemes have been introduced so far and they are explored as well by many researchers. Graph labelings have enormous applications within mathematics as well as to several areas of computer science and communication networks. Various applications of graph labeling are reported in the work of Yegnanaryanan and Vaidhyanathan [3]. For a dynamic survey on various graph labeling problems along with an extensive bibliography we refer to Gallian [4]. The concept of prime labeling has attracted many researchers as the study of prime numbers is of great importance because prime numbers are scattered and there are arbitrarily large gaps in the sequence of prime numbers. Vaidya and Prajapati [7,8] have investigated many results on prime labeling. Same authors [9] have discussed prime labeling in the context of duplication of graph elements. Motivated through the concepts of prime labeling and cordial labeling, a new concept termed as a prime cordial labeling was introduced by Sundaram et al. [10] which contains blend of both the labelings.
then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits prime cordial labeling is called a prime cordial graph.
Many graphs are proved to be prime cordial in the work of Sundaram et al. [10]. Prime cordial labeling for some cycle related graphs has been discussed by Vaidya and Vihol [11]. Prime cordial labeling in the context of some graph operations has been discussed by Vaidya and Vihol [12] and Vaidya and Shah [13,14]. Vaidya and Shah [14] have proved that the wheel graph admits prime cordial labeling for ≥ 8 while same authors in [15] have discussed prime cordial labeling for some wheel related graphs. Babitha and Baskar Babujee [16] have exhibited prime cordial labeling for some cycle related graphs and discussed the duality of prime cordial labeling. The same authors in [17] have derived some characterizations of prime cordial graphs and investigated various methods to construct larger prime cordial graphs using existing prime cordial graphs. We investigate a method different from existing one to construct larger prime cordial graph from an existing prime cordial graph.

Definition 6. The wheel
is defined to be the join 1 + . The vertex corresponding to 1 is known as apex and vertices corresponding to cycle are known as rim vertices while the edges corresponding to cycle are known as rim edges.
Definition 7. The bistar , is a graph obtained by joining the center (apex) vertices of two copies of 1, by an edge.  Definition 11 (see [18]). Let = ( ( ), ( )) be a graph with = 1 ∪ 2 ∪ 3 ∪ ⋅ ⋅ ⋅ ∪ , where each is a set of vertices having at least two vertices of the same degree and = \ ∪ =1 . The degree splitting graph of G denoted by ( ) is obtained from by adding vertices 1 , 2 , 3 , . . . , and joining to each vertex of for 1 ≤ ≤ .

Case (i) ( is even and
is of any size ). Since is a prime cordial graph, we assign vertex labels such that ( ) = 1 ( ), where ∈ ( ) = ( 1 ) ∩ ( ) and ≤ ≤ : Since is even and 1 ( 1 ) = 2 and 1 ( 2 ) = 4, 1 and 2 are adjacent to each , ≤ ≤ . And this vertex assignment generates edges with label 1 and edges with label 0. Following Table 1 gives edge condition for prime cordial labeling for 1 under .
Since is a prime cordial graph, we keep the vertex label of all the vertices of in 1 as it is. Therefore ( ) = 1 ( ), where ∈ ( ) and ≤ ≤ : Since is odd and 1 ( 1 ) = 2 and 1 ( 2 ) = 4, 1 and 2 are adjacent to each , ≤ ≤ . And this vertex assignment generates + 1 edges with label 0 and − 1 edges with label 1. Table 1 Edge conditions for Edge conditions for 1 Case (iii) ( is odd, is even, and is odd with Here is even, is odd, and is a prime cordial graph with Since is a prime cordial graph, we keep the vertex label of all the vertices of in 1 as it is. Therefore ( ) = 1 ( ), where ∈ ( ) and ≤ ≤ : Since is odd and 1 ( 1 ) = 2 and 1 ( 2 ) = 4, 1 and 2 are adjacent to each , ≤ ≤ . And this vertex assignment generates − 1 edges with label 0 and + 1 edges with label 1. Therefore edge conditions for 1 under are (0) = ⌈ /2⌉ + − 1 and (1) = ⌊ /2⌋ + + 1. Therefore, Hence, in all the cases discussed above, 1 admits prime cordial labeling. Case 1 ( = 3 to 7 and = 9). In order to satisfy the edge condition for prime cordial labeling in DF 3 it is essential to label four edges with label 0 and four edges with label 1 out of eight edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most one edge and 1 label for at least seven edges. That is, | (0) − (1)| = 6 > 1. Hence, DF 3 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF 4 it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most three edges and 1 label for at least eight edges. That is, | (0) − (1)| = 5 > 1. Hence, DF 4 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF 5 it is essential to label seven edges with label 0 and seven edges with label 1 out of fourteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least ten edges. That is, | (0) − (1)| = 6 > 1. Hence, DF 5 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF 6 it is essential to label eight edges with label 0 and nine edges with label 1 out of seventeen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least eleven edges. That is, | (0) − (1)| = 5 > 1. Hence, DF 6 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF 7 it is essential to label ten edges with label 0 and ten edges with label 1 out of twenty edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most eight edges and 1 label for at least twelve edges. That is, | (0) − (1)| = 4 > 1. Hence, DF 7 is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in DF 9 it is essential to label thirteen edges with label 0 and thirteen edges with label 1 out of twenty-six edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most twelve edges and 1 label for at least fourteen edges. That is, | (0) − (1)| = 2 > 1. Hence, DF 9 is not a prime cordial graph. Now for the remaining three cases let 3 = largest even number ≤ +2, and 4 = largest odd number ≤ + 2.   ( 2 ) = 6.
Hence, DF is a prime cordial graph for = 8 and ≥ 10.
Case 1 ( = 3, 4, 5). For = 3, to satisfy the edge condition for prime cordial labeling, it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least seven edges. That is, | (0) − (1)| = 3 > 1. Hence, for = 3, is not a prime cordial graph. For = 4, to satisfy the edge condition for prime cordial labeling, it is essential to label six edges with label 0 and seven edges with label 1 out of thirteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least nine edges. That is, | (0) − (1)| = 5 > 1. Hence, for = 4, is not a prime cordial graph. For = 5, to satisfy the edge condition for prime cordial labeling it is essential to label seven edges with label 0 and eight edges with label 1 out of fifteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least nine edges. That is, | (0) − (1)| = 3 > 1. Hence, for = 5, is not a prime cordial graph.    Proof. Let V 0 be the apex vertex of and let V 1 , V 2 , . . . , V be the rim vertices. Without loss of generality we duplicate the spoke edge = V 0 V 1 by an edge = 1 2 and call the resultant graph . Then | ( )| = + 3 and | ( )| = 3 + 2. To define : ( ) → {1, 2, 3, . . . , + 3}, we consider following three cases.
Case 1 ( = 3 to 6 and = 8). For = 3, to satisfy the edge condition for prime cordial labeling it is essential to label five edges with label 0 and six edges with label 1 out of eleven edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least seven edges. That is, | (0) − (1)| = 3 > 1. Hence, for = 3, is not a prime cordial graph. For = 4, to satisfy the edge condition for prime cordial labeling, it is essential to label seven edges with label 0 and seven edges with label 1 out of fourteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most four edges and 1 label for at least ten edges. That is, | (0) − (1)| = 6 > 1. Hence, for = 4, is not a prime cordial graph. For = 5, to satisfy the edge condition for prime cordial labeling, it is essential to label eight edges with label 0 and nine edges with label 1 out of seventeen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 label for at least eleven edges. That is, | (0) − (1)| = 5 > 1. Hence, for = 5, is not a prime cordial graph. For = 6, to satisfy the edge condition for prime cordial labeling, it is essential to label ten edges with label 0 and ten edges with label 1 out of twenty edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most eight edges and 1 label for at least twelve edges. That is, | (0) − (1)| = 4 > 1. Hence, for = 6, is not a prime cordial graph. For = 8, to satisfy the edge condition for prime cordial labeling, it is essential to label thirteen edges with label 0 and thirteen edges with label 1 out of twenty-six edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most twelve edges and 1 label for at least fourteen edges. That is, | (0) − (1)| = 2 > 1. Hence, for = 8, is not a prime cordial graph.
Hence, is a prime cordial graph for = 7 and ≥ 9.   odd.
Hence, DS( ) is a prime cordial graph.
Illustration 7. Prime cordial labeling of the graph DS( 5,5 ) is shown in Figure 6.

Conclusion
A new approach for constructing larger prime cordial graph from the existing prime cordial graph is investigated.