b-coloring of Cartesian product of odd graphs

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. The b-spectrum S-b(G) of a graph G is the set of positive integers k, chi(G) <= k <= b(G), for which G has a b-coloring using k colors. A graph G is b-continuous if S-b(G)= {chi(G),..., b(G)}. It is known that for any two graphs G and H, b(G square H) >= max{b(G), b(H)}, where square stands for the Cartesian product. In this paper, we determine some families of graphs G and H for which b(G square H) >= b(G) b(H) - 1. Further we show that if O-ki, i = 1,2,...,n are odd graphs with k(i) >= 4 for each i, then O-k1 square O-k2 square... square O-kn is b-continuous and b(O-k1 square O-k2 square ... square O-kn) = 1 + Sigma(n)(i=1) k(i).