$b$-Chromatic number of some wheel related graphs

Abstract

A proper coloring $f$ is a $b$-coloring of the vertices of graph $G$ such that in each color class there exists a vertex that has neighbours in every other color classes. The $b$-chromatic number $\varphi(G)$ of a graph $G$ is the largest integer $k$ for which $G$ admits a $b$-coloring with $k$ colors. If $\chi(G)$ is the chromatic number of $G$ and $b$-coloring exists for every integer $k$ satisfying the inequality $\chi(G) \leq k \leq \varphi(G)$ then $G$ is called $b$-continuous. The $b$-spectrum $S_b(G)$ of a graph $G$ is the set of $k$ integers(colors) for which $G$ has a $b$-coloring. We investigate $b$-chromatic number for the graphs obtained from wheel $W_n$ by means of duplication of vertices. We also discuss $b$-continuity and $b$-spectrum for such graphs.