The b-chromatic number and f-chromatic vertex number of regular graphs

The b-chromatic number of a graph G, denoted by b(G), is the largest positive integer k such that there exists a proper coloring for G with k colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The f -chromatic vertex number of a d-regular graph G, denoted by f(G), is the maximum number of dominant vertices of distinct colors in a proper coloring with d+1 colors. El Sahili and Kouider conjectured that b(G) = d+1 for any d-regular graph G of girth 5. We study this conjecture by giving some partial answers under supplementary conditions.