Acyclic colourings of graphs with obstructions
Résumé
Given a graph $G$, a colouring of $G$ is acyclic if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $χ_a(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$ with no bicoloured cycle. In general, when $G$ has maximum degree $Δ$, it is known that $χ_a(G) = O(Δ^{4/3})$ as $Δ\to \infty$. We study the effect on this bound of further requiring that $G$ does not contain some fixed subgraph $F$ on $t$ vertices. We establish that the bound is constant if $F$ is a subdivided tree, $O(t^{8/3}Δ^{2/3})$ if $F$ is a forest, $O(\sqrt{t}Δ)$ if $F$ is bipartite and 1-acyclic, $2Δ+ o(Δ)$ if $F$ is an even cycle of length at least $6$, and $O(t^{1/4}Δ^{5/4})$ if $F=K_{3,t}$.