ON THE b-CHROMATIC NUMBER OF SOME GRAPH PRODUCTS

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer phi(G) for which a graph has a b-coloring with phi(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G boxed times H, the lexicographic product G[H] and the direct product G x H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of P-n boxed times H, C-n boxed times H, P-n[H], C-n[H], and K-m,K-n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.