$b$-chromatic Number for the Graphs Obtained by Duplicating Edges

Abstract

A $b$-colouring of a graph $G$ is a proper vertex colouring of $G$ such that each colour class contains a vertex that has atleast one neighbour in every other colour class and $b$-chromatic number of a graph $G$ is the largest integer $\phi(G)$ for which $G$ has a $b$-colouring with $\phi(G)$ colours. In this paper, we have obtained the $b$-chromatic number of the graphs $E_n, F_n$ and the graphs obtained by duplicating all the edges of path, cycle, complete graph, wheel graph, Ladder graph $L_n$ by vertices.