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

The Grundy and b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G, denoted by Gamma(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 Gamma(G) - inverted right perpendicularlog Gamma(G)inverted right perpendicular <= b(G), if the girth of G is sufficiently large with respect to its maximum degree. Next, we prove that if G is K-2,K-3-free then Gamma(G) <= (b(G))(3)/2. These results confirm a previous conjecture for these families of graphs. (C) 2021 Elsevier B.V. All rights reserved.