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

TL;DR

A comparative study of the Grundy and the b-chromatic number of graphs, which have been studied widely but separately in the literature, is presented.

Abstract

The Grundy and the {\rm 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 {\rm b}-chromatic number ${\rm{b}}(G)$ 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 $\{G_n\}_{n\geq 1}$ with limited {\rm b}-chromatic number but $Γ(G_n)\rightarrow \infty$. We obtain families of graphs $\mathcal{F}$ such that for some adequate function $f(.)$, $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. This verifies a previous conjecture for these families.

Figures (5)

• Figure 1: $T_2, T_3, T_4, T_5$, from left to right
• Figure 2: A cactus graph which is not b-monotone.
• Figure 3: Recoloring process in cactus graphs (related to Theorem \ref{['bound-cactus']})
• Figure 4: A situation in the proof of Theorem \ref{['K4e']}
• Figure 5: An (almost) sharpness example for Theorem \ref{['K4e']}