Acyclic Choosability of Graphs with Bounded Degree

An acyclic colouring of a graph G is a proper vertex colouring such that every cycle uses at least three colours. For a list assignment L = { L ( v )∼ v ∈ V ( G )}, if there exists an acyclic colouring ρ such that ρ ( v ) ∈ L ( v ) for each v ∈ V ( G ), then ρ is called an acyclic L -list colouring of G . If there exists an acyclic L -list colouring of G for any L with ∣ L ( v )∣> k for each v ∈ V ( G ), then G is called acyclically k -choosable. In this paper, we prove that every graph with maximum degree Δ ≤ 7 is acyclically 13-choosable. This upper bound is first proposed. We also make a more compact proof of the result that every graph with maximum degree Δ ≤ 3 (resp., Δ ≤ 4) is acyclically 4-choosable (resp., 5-choosable).