A new approach to b-coloring of regular graphs

Let G be a graph and c:V(G)→{1,…,k} a proper k-coloring of G, i.e., c(u)≠c(v) for every edge uv from G. A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in its closed neighborhood. The maximum number of colors k admitting b-coloring of G is the b-chromatic number χb(G).We present two separate approaches to the conjecture posed by Blidia et al. (2009) [2] that χb(G)=d+1 for every d-regular graph of girth at least five except the Petersen graph.