Acyclic colourings of graphs with obstructions
Quentin Chuet, Johanne Cohen, François Pirot
TL;DR
The paper studies acyclic colourings and the acyclic chromatic number χ_a in graphs with maximum degree Δ that avoid a fixed obstruction F on t vertices. It introduces a general framework (Γ-proper, Π-acyclic colourings) and a main counting theorem that yields lower bounds on the number of colourings, which in turn leads to broad upper bounds a(d, F) for various obstructions, including trees, certain bipartite obstructions, even cycles, and subdivisions like ⟨K_{s,t}^{1/k}⟩. It pairs these upper bounds with lower bounds derived from random graphs G(n,p) and G(n,n,p) to map out several asymptotic regimes, showing that the extremal a(d, F) can exhibit a rich variety of behaviors depending on the structure of F, with some obstructions forcing near-linear growth in Δ and others keeping χ_a bounded or sublinear. The paper further extends the analysis to cycle obstructions, providing sharp constants in several cases (notably C_4-free graphs and girth7 regimes) and culminates in a detailed landscape of χ_a under F-free constraints, along with open problems and directions for future work.
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 $χ_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of $G$ exists. When $G$ has maximum degree $Δ$, it is known that $χ_a(G) = \mathcal {O}(Δ^{4/3})$ as $Δ\to \infty$, and that $χ_a(G) = \mathcal {O}(\sqrt{t} \cdot Δ)$ 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 $Δ$ that do not contain some fixed subgraph $F$ on $t$ vertices. We establish that this extremal value is at most $\mathcal {O}(t^{8/3}Δ^{2/3})$ if $F$ is a tree, $\mathcal {O}(\sqrt{t} \cdot Δ)$ if $F$ is bipartite and can be made acyclic with the removal of one vertex, $2Δ+ \mathcal {O}(tΔ^{2/3})$ if $F$ is an even cycle of length at least $6$, and $\mathcal {O}(t^{1/4}Δ^{5/4})$ 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 $Δ^{\mathcal{O}(1/t)}$ factor.