Acyclic colourings of graphs with obstructions

TL;DR

This paper investigates the maximum acyclic chromatic number of graphs with bounded degree that exclude certain subgraphs, establishing upper bounds for various obstructions and analyzing their tightness through probabilistic methods.

Contribution

It provides new upper bounds on the acyclic chromatic number for graphs excluding specific subgraphs, and characterizes how these bounds vary with different obstructions.

Findings

Upper bounds for acyclic chromatic number depending on the forbidden subgraph

Analysis of extremal behavior for various obstructions

Lower bounds from random graph models matching upper bounds up to polylog factors

Abstract

Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $\chi_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of $G$ exists. When $G$ has maximum degree $\Delta$, it is known that $\chi_a(G) = \mathcal {O}(\Delta^{4/3})$ as $\Delta \to \infty$, and that $\chi_a(G) = \mathcal {O}(\sqrt{t} \cdot \Delta)$ if in addition $G$ does not contain $K_{2,t}$ as a subgraph. We study the extremal value of the acyclic chromatic number in the class of graphs of maximum degree $\Delta$ that do not contain some fixed subgraph $F$ on $t$ vertices. We establish that this extremal value is at most $\mathcal {O}(t^{8/3}\Delta^{2/3})$ if $F$ is a tree, $\mathcal {O}(\sqrt{t} \cdot \Delta)$ if $F$ is bipartite and can be made acyclic with the removal of one vertex, $2\Delta + \mathcal {O}(t\Delta^{2/3})$ if $F$ is an even cycle of length at least $6$, and $\mathcal {O}(t^{1/4}\Delta^{5/4})$ if $F=K_{3,t}$. Moreover, we exhibit an infinite family of obstructions $F$ that each induces a different asymptotic behaviour for this extremal value. This is obtained with the derivation of lower bounds that come from the analysis of the acyclic chromatic number of a random graph drawn from either $G(n,p)$ or $G(n,n,p)$, that we entirely determine up to a ${\rm polylog}(n)$ factor. As a byproduct, we can certify that most of our results are tight up to a $\Delta^{\mathcal{O}(1/t)}$ factor.