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 $$\Gamma (G)$$ is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number $$\mathrm{{b}}(G)$$ 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 $$\{G_n\}_{n\ge 1}$$ with limited b-chromatic number but $$\Gamma (G_n)\rightarrow \infty $$ . We obtain families of graphs $${\mathcal {F}}$$ such that for some adequate function f(.), $$\Gamma (G)\le f(\mathrm{{b}}(G))$$ , for each graph G from the family. This verifies a previous conjecture for these families.