b-continuity and the lexicographic product of graphs

A k-coloring c of a graph G=(V,E) is a b-coloring if for every color class ci, 1≤i≤k, there is a vertex colored i whose neighborhood intersects every other color class cj of c. The b-chromatic number of G,χb(G), is the greatest k such that G admits a b-coloring with k colors. Every optimal coloring of G is a b-coloring. Therefore χ(G)≤χb(G). G is b-continuous if for every k,χ(G)≤k≤χb(G),G admits a b-coloring with k colors. In this paper, we are interested in b-continuous graphs G[H] which are the lexicographic product of two b-continuous graphs G and H. We give partial results on the spectrum of G[H] and we examine its b-continuity for specific classes of G and H.