On Grundy and b-chromatic number of some families of graphs: a comparative study

Abstract

The Grundy and the 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. This paper presents a comparative study of these coloring parameters. There exists a sequence {G n } n≥1 with limited b-chromatic number but Γ(G n ) → ∞. We obtain families of graphs F such that for some adequate function f (.), Γ(G) ≤ f (b(G)), for each graph G from the family. This verifies a previous conjecture for these families.