Note on b-colorings in Harary graphs

Abstract

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. The b-chromatic number b(G) is the largest integer k such that G admits a b-coloring with k colors. In this note, according to the values taken by the order n of a graph, we determine exact values or bounds for the b-chromatic number of H2m;n which is the Harary graph Hk;n when k is even: Therefore our result improves the result concerning the b-chromatic of p-th power graphs of cycles and give a negative answer to the open problem of E¤antin and Kheddouci