An Improved Bound of Acyclic Vertex-Coloring

TL;DR

This paper improves the upper bound on the acyclic chromatic number of graphs with large maximum degree, showing it can be bounded by a function involving ^{4/3} with a smaller constant than previous results.

Contribution

The paper establishes a tighter upper bound on the acyclic chromatic number for graphs with sufficiently large maximum degree, refining earlier bounds.

Findings

New bound: _{ ext{acyclic}} \u2264 eil /3 + \u001 + 1 for large .

Improved constant factor over previous bound (3/2) ^{4/3}.

Bound holds for all  > 2^{-1/3} with sufficiently large .

Abstract

The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $\alpha>2^{-1/3}$ there exists an integer $\Delta_{\alpha}$ such that if the maximum degree $\Delta$ of a graph is at least $\Delta_{\alpha}$, then the acyclic chromatic number of the graph is at most $\lceil\alpha {\Delta}^{4/3} \rceil +\Delta+ 1$. The previous best bound, due to Gon\c{c}alves et al (2020), was $(3/2) \Delta^{4/3} + O(\Delta)$.