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 χ b ( G ) for which G has a b-coloring with χ 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 [ χ ( G ) , χ 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 χ b ( G [ H ] ) , namely χ b ( G [ K t ] ) , where t = χ b ( H ) , and prove that if G [ K ℓ ] 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 ] ) , χ b ( G [ K t ] ) ] . We also prove that G [ K ℓ ] is b-continuous, for every positive integer ℓ , whenever G is a P 4 -sparse graph, and we give further results on the b-spectrum of G [ K ℓ ] , when G is chordal. Finally, we determine the value of χ b ( T [ K ℓ ] ) , when T is a tree.