  Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent
vertices u and v have different colors c(u) and c(v). 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. We present two separate approaches
to the conjecture posed by Blidia et. al that the b-chromatic number equals to
d+1 for every d-regular graph of girth at least five except the Petersen graph.
