\(b\)-Chromatic Number of Triple Star Graph Families

Abstract

A $b$-coloring of a graph \(G\) is a proper coloring of the vertices of \(G\) such that there exists a vertex in each color class joined to atleast a vertex in each other color class, such a vertex is called a dominating vertex. The \(b\)-chromatic number of a graph \(G\), denoted by \(b(G)\), is the maximal integer \(k\) such that \(G\) may have a \(b\)-coloring by \(k\) colors. In this paper, we investigate the \(b\)-chromatic number of Central graph, Middle graph, Total graph and Line graph of Triple Star graph, denoted by \(C(K_{1,n,n,n})\), \(M(K_{1,n,n,n})\), \(T(K_{1,n,n,n})\) and \(L(K_{1,n,n,n})\), respectively.