Some comparative results concerning the Grundy and b-chromatic number of graphs

Abstract

The Grundy and b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G, denoted by Γ(G) is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number b(G) is the maximum number of colors used in any color-dominating coloring of G. Because the nature of these colorings are different they have been studied widely but separately in the literature. In this paper we first prove that Γ(G)−⌈logΓ(G)⌉≤b(G), if the girth of G is sufficiently large with respect to its maximum degree. Next, we prove that if G is K2,3-free then Γ(G)≤(b(G))3/2. These results confirm a previous conjecture for these families of graphs.