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)similar to v is an element of V(G)}, if there exists an acyclic colouring rho such that rho(v) is an element of L(v) for each v is an element of V(G), then rho is called an acyclic L-list colouring of G. If there exists an acyclic L-list colouring of G for any L with divide L(v) divide > k for each v is an element of V (G), then G is called acyclically k-choosable. In this paper, we prove that every graph with maximum degree Delta <= 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 Delta <= 3 (resp., Delta <= 4) is acyclically 4-choosable (resp., 5-choosable).