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.

翻译：给定图 $G$，若 $G$ 的一种着色既是正常着色，且每个圈至少包含三种颜色，则称该着色为\emph{无环}着色。其无环色数 $χ_a(G)$ 是使得 $G$ 存在无环 $k$ 着色的最小整数 $k$。已知当 $G$ 的最大度为 $Δ$ 时，随着 $Δ\to \infty$，有 $χ_a(G) = \mathcal {O}(Δ^{4/3})$；若进一步假设 $G$ 不包含 $K_{2,t}$ 作为子图，则有 $χ_a(G) = \mathcal {O}(\sqrt{t} \cdot Δ)$。本文研究在最大度为 $Δ$ 且不包含某个具有 $t$ 个顶点的固定子图 $F$ 的图类中，无环色数的极值。我们证明：若 $F$ 是树，则该极值至多为 $\mathcal {O}(t^{8/3}Δ^{2/3})$；若 $F$ 是二分图且可通过删除一个顶点变为无环图，则该极值至多为 $\mathcal {O}(\sqrt{t} \cdot Δ)$；若 $F$ 是长度至少为 $6$ 的偶圈，则该极值至多为 $2Δ+ \mathcal {O}(tΔ^{2/3})$；若 $F=K_{3,t}$，则该极值至多为 $\mathcal {O}(t^{1/4}Δ^{5/4})$。此外，我们构造了一个无限禁图族 $F$，其中每个禁图都诱导出该极值的一种不同的渐近行为。这些结论的获得源于对从 $G(n,p)$ 或 $G(n,n,p)$ 中抽取的随机图的无环色数下界的分析，我们完全确定了该下界（相差一个 ${\rm polylog}(n)$ 因子）。作为副产品，我们可以证明我们的大部分结果在相差一个 $Δ^{\mathcal{O}(1/t)}$ 因子的意义下是紧的。