Research Article  Open Access
Some New Results on Prime Cordial Labeling
Abstract
A prime cordial labeling of a graph with the vertex set is a bijection such that each edge is assigned the label 1 if and 0 if ; 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 . 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 wheel admits prime cordial labeling.
1. 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].
Definition 2. A labeling is called binary vertex labeling of and is called the label of the vertex of under .
Notation 1. If for an edge , the induced edge labeling is given by . Then
Definition 3. A binary vertex labeling of a graph is called a cordial labeling if and . A graph is cordial if it admits cordial labeling.
The concept of cordial labeling was introduced by Cahit [5].
The notion of prime labeling was originated by Entringer and was introduced by Tout et al. [6].
Definition 4. A prime labeling of a graph is an injective function such that for, every pair of adjacent vertices and , . The graph which admits a prime labeling is called a prime graph.
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.
Definition 5. A prime cordial labeling of a graph with vertex set is a bijection and if the induced function is defined by 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 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 . The vertex corresponding to 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 by an edge.
Definition 8. The fan is the graph obtained by taking concurrent chords in cycle . The vertex at which all the chords are concurrent is called the apex vertex. In other words, .
Definition 9. The double fan consists of two fan graphs that have a common path. In other words, .
Definition 10. The duplication of an edge of graph produces a new graph by adding an edge such that and .
Definition 11 (see [18]). Let be a graph with , where each is a set of vertices having at least two vertices of the same degree and . The degree splitting graph of G denoted by is obtained from by adding vertices and joining to each vertex of for .
2. Main Results
Theorem 12. Let with be a prime cordial graph and let be a bipartite graph with bipartition with and . If is the graph obtained by identifying the vertices and of with the vertices of having labels 2 and 4, respectively, then admits prime cordial labeling in any of the following cases:(i)is even and is of any size ;(ii) are odd with ;(iii) is odd, is even, and is odd with .
Proof. Let be a prime cordial graph and let be the prime cordial labeling of . Let be the vertices of such that and . Consider the with bipartition with and . Now identify the vertices to and to and denote the resultant graph as . Then and so and . To define , we consider the following three cases.
Case (i) ( is even and is of any size ). Since is a prime cordial graph, we assign vertex labels such that , where and :
Since is even and and , and 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 under .
From Table 1, we have .
Case (ii) ( are odd with ). Here and both are 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 as it is. Therefore , where and :
Since is odd and and , and are adjacent to each , . And this vertex assignment generates edges with label 0 and edges with label 1.
Therefore edge conditions for under are and . Therefore, . Hence, for graph .
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 as it is. Therefore , where and :
Since is odd and and , and are adjacent to each , . And this vertex assignment generates edges with label 0 and edges with label 1.
Therefore edge conditions for under are and . Therefore, . Hence, for graph .
Hence, in all the cases discussed above, admits prime cordial labeling.

Illustrationâ€‰â€‰1. Consider the graph as shown in Figure 1, with and . is a prime cordial graph with . Take and construct graph . In accordance with Case (ii) of Theorem 12, a prime cordial labeling of is as shown in Figure 1. Here .
(a)
(b)
Theorem 13. Double fan is a prime cordial graph for and .
Proof. Let be the double fan with apex vertices and are vertices common path. Then and . To define , we consider the following five cases.
Case 1â€‰ ( to 7 and ). In order to satisfy the edge condition for prime cordial labeling in 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, . Hence, is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in 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, . Hence, is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in 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, . Hence, is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in 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, . Hence, is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in 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, . Hence, is not a prime cordial graph.
In order to satisfy the edge condition for prime cordial labeling in it is essential to label thirteen edges with label 0 and thirteen edges with label 1 out of twentysix 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, . Hence, is not a prime cordial graph.
Case 2â€‰ (). For , , and , , , , , , , and . Then , .
For , , and , , , , , , , , and . Then , .
For , , and , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , and . Then , .
Now for the remaining three cases let
largest even number , and largest odd number .
Case 3. Consider
For the vertices we assign the vertex labels in the following order: , , , , , , , , , , , , , , , , .
Case 4. Consider
Let . Consider
Now for remaining vertices assign the labels ,, all the odd numbers in ascending order.
Caseâ€‰â€‰5. Let .
SubCase 1. is even. Consider
For the vertices we assign the vertex labels in the following order: , , , , , , , remaining odd numbers in ascending order.
SubCase 2. is odd. Consider
For the vertices we assign the vertex labels in the following order: , , , , , , , , remaining odd numbers in ascending order.
In view of the above defined labeling pattern for Cases 3, 4, and 5, we have
Thus, we have .
Hence, is a prime cordial graph for and .
Illustrationâ€‰â€‰2. For the graph , and . In accordance with Theorem 13 we have , , , and . Here so labeling pattern described in Case 4 will be applicable and . The corresponding prime cordial labeling is shown in Figure 2. Here .
Illustrationâ€‰â€‰3. For the graph , and . In accordance with Theorem 13, we have , , , and . Here and so labeling pattern described in SubCase of Case will be applicable and . And corresponding labeling pattern is as below: For the vertices we assign the vertex labels 39, 38, 37, 36, 35, 34, 32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10, 8, 4, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, and 33, respectively, where . Therefore is a prime cordial graph.
Theorem 14. The graph obtained by duplication of an arbitrary rim edge by an edge in is a prime cordial graph, where .
Proof. Let be the apex vertex of and let be the rim vertices. Without loss of generality we duplicate the rim edge by an edge and call the resultant graph as . Then and . To define , we consider the following four cases.
Case 1. For , 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, . Hence, for , is not a prime cordial graph.
For , 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, . Hence, for , is not a prime cordial graph.
For , 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, . Hence, for , is not a prime cordial graph.
Case 2 to 10). For , , and , , , , , , and . Then , .
For , , and , , , , , , , and . Then , .
For , , and , , , , , , , , and . Then , .
For , , and , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , and . Then , .
Case 3 ( is even, . Consider
In view of the above defined labeling pattern for Case 3, we have and for and and for .
Case 4 ( is odd, ). Consider
In view of the above defined labeling pattern for Caseâ€‰â€‰4, we have and for and and for .
In view of Cases 2 to 4 we have .
Hence, is a prime cordial graph for .
Illustrationâ€‰â€‰4. Let be the graph obtained by duplication of an arbitrary rim edge by an edge in wheel . Then and . In accordance with Theorem 14, Caseâ€‰â€‰4 will be applicable and the corresponding prime cordial labeling is shown in Figure 3. Here , .
Theorem 15. The graph obtained by duplication of an arbitrary spoke edge by an edge in wheel is prime cordial graph, where and .
Proof. Let be the apex vertex of and let be the rim vertices. Without loss of generality we duplicate the spoke edge by an edge and call the resultant graph . Then and . To define , we consider following three cases.
Case 1 to 6 and . For , 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, . Hence, for , is not a prime cordial graph.
For , 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, . Hence, for , is not a prime cordial graph.
For , 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, . Hence, for , is not a prime cordial graph.
For , 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, . Hence, for , is not a prime cordial graph.
For , 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 twentysix 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, . Hence, for , is not a prime cordial graph.
Case 2, to 12, and .â€‰â€‰For , , and , , , , , , , and . Then , .
For , , and , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , , , , , , and . Then , .
For , , and , , , , , , , , ,, , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , , , , , , , , , , and . Then , .
For , , and , , , , , , , , , , , , , , , , , , , , , , and . Then , .
For the next case let , , , , ,
Case 3â€‰ ( ( and )). Consider
For ,
In view of the above defined labeling pattern for Case 3, we have and .
Thus for Casesâ€‰â€‰2 andâ€‰â€‰3 we have .
Hence, is a prime cordial graph for and .
Illustrationâ€‰â€‰5. Let be the graph obtained by duplication of arbitrary spoke edge by an edge of wheel . Then and . In accordance with Theorem 15 we have , , , , , and . Here so labeling pattern described in Case 3 will be applicable. The corresponding prime cordial labeling is shown in Figure 4. It is easy to visualise that , .
Theorem 16. DS is a prime cordial graph.
Proof. Consider with . Here , where and . Now in order to obtain from , we add corresponding to . Then and so . We define vertex labeling as follows.
Let be the highest prime number and . Consider
For ,
And for vertices we assign distinct odd numbers () in ascending order starting from 5.
In view of the above defined labeling pattern, if is even number, then , ; otherwise , .
Thus, .
Hence, is a prime cordial graph.
Illustrationâ€‰â€‰6. Prime cordial labeling of the graph is shown in Figure 5.
Theorem 17. DS is a prime cordial graph.
Proof. Consider with , where are pendant vertices. Here , where and . Now in order to obtain from , we add corresponding to . Then and