Bounds for the Grundy chromatic number of graphs in terms of domination number

TL;DR

This paper establishes new upper bounds for the Grundy chromatic number of graphs based on domination number and girth, using star partitions, addressing an NP-complete problem.

Contribution

It introduces the first bounds for the Grundy number in terms of domination number using star partitions and explores bounds involving girth.

Findings

Upper bounds for Grundy number via domination number

Bounds involving girth and domination number

Use of star partitions for bounding Grundy number

Abstract

For any graph $G$, the Grundy (or First-Fit) chromatic number of $G$, denoted by $\Gamma(G)$ (also $\chi_{_{\sf FF}}(G)$), is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. Determining the Grundy number is $NP$-complete, and obtaining bounds for $\Gamma(G)$ in terms of the known graph parameters is an active research topic. By a star partition of $G$ we mean any partition of $V(G)$ into say $V_1, \ldots, V_k$ such that each $G[V_i]$ contains a vertex adjacent to any other vertex in $V_i$. In this paper using the star partition of graphs we obtain the first upper bounds for the Grundy number in terms of the domination number. We also prove some bounds in terms of the domination number and girth of graphs.