The lexicographic product of some chordal graphs and of cographs preserves b-continuity

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, and also that the cartesian product and the strong product do not preserve b-continuity. However, the same is not known to be true about the lexicographic product G[H]. Here, we prove that G[H] is b-continuous whenever H is b-continuous and G is an interval graph, a block graph or a cograph.