Study of b-Chromatic Number of Wheel Graph

Abstract

In this paper we have generalized some of basic result on chromatic number.The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. All graph considered here are simple, undirected and finite. For a graph G, we denote by V(G)its vertex set and by E(G)its edge set;|V(G)|is the order and x(G)is the chromatic number of G. for a graph G and a vertex x of G. Let G=(V,E) be an undirected and loopless graph.The b-chromatic number of a graph G is the largest iteger k such that G admits a proper k-colouring in which every colour class contains atleast one vertex adjacent to some vertex in all the other colour classes. A proper k-colouring of a graph G=(V(G),E(G)) is a mapping f:V(G)->N such that every two adjacent vertices receive different colors. The chromatic number of a graph G is denoted by X(G), is the minimum number foe which G has a proper k-colouring. The set of vertices with a specific colour is called a colour class. The b-chromatic number ?(G)is the largest integer k such that G admits a b-colouring with k colour.