On the b-Continuity of the Lexicographic Product of Graphs

Abstract

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 $$\chi _b(G)$$ for which G has a b-coloring with $$\chi _b(G)$$ colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval $$[\chi (G),\chi _b(G)]$$ . 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 $$\chi _b(G[H])$$ , namely $$\chi _b(G[K_t])$$ , where $$t=\chi _b(H)$$ , and prove that if $$G[K_\ell ]$$ is b-continuous for every positive integer $$\ell $$ , then G[H] admits a b-coloring with k colors, for every k in the interval $$[\chi (G[H]),\chi _b(G[K_t])]$$ . We also prove that $$G[K_\ell ]$$ is b-continuous, for every positive integer $$\ell $$ , whenever G is a $$P_4$$ -sparse graph, and we give further results on the b-spectrum of $$G[K_\ell ]$$ , when G is chordal. Finally, we determine the value of $$\chi _b(T[K_\ell ])$$ , when T is a tree.