# Acyclic colourings of graphs with obstructions

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→∞. 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^8/3Δ^2/3) if F is a forest, O(√(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^1/4Δ^5/4) if F=K_3,t.

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