Some New Results on Prime Cordial Labeling

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