On the b-chromatic number of regular bounded graphs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The b-chromatic number of a graph G, denoted by b(G), is the maximum integer k such that G admits a b-coloring with k colors. El Sahili and Kouider conjectured that b(G) = d + 1 for d-regular graph with girth 5, d ≥ 4. In this paper, we prove that this conjecture holds for d-regular graph with at least d + d vertices. More precisely we show that b(G) = d+1 for d-regular graph with at least d + d vertices and containing no cycle of order 4. We also prove that b(G) = d+1 for d-regular graphs with at least 2d+2d−2d vertices improving Cabello and Jakovac bound.