On the b-continuity of the lexicographic product of graphs

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer χb(G) for which G has a b-coloring with χb(G) colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval [χ(G), χb(G)]. It is known that not all graphs are b-continuous. Here, we investigate whether the lexicographic product G[H ] of b-continuous graphs G and H is also b-continuous. Using homomorphisms, we provide a new lower bound for χb(G[H ]), namely χb(G[Kt]), where t = χb(H), and prove that if G[Kl] is b-continuous for every positive integer l, then G[H ] admits a b-coloring with k colors, for every k in the interval [χ(G[H ]), χb(G[Kt])]. We also prove that G[Kl] is b-continuous, for every positive integer l, whenever G a P4-sparse graph, and we give further results on the b-spectrum of G[Kl], when G is chordal.