Acyclic choosability of graphs with small maximum degree

A proper vertex coloring of a graph C = (V, E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable if for a given list assignment L = {L(v) : v epsilon V}, there exists a proper coloring c of G such that c(v) epsilon L(v) for all v epsilon V. If G is L-list colorable for every list assignment with vertical bar L(v)vertical bar >= k for all v epsilon V, then G is said k-choosable. A graph is said to be acyclically k-choosable if the coloring obtained is acyclic. In this paper, we study the acyclic choosability of graphs with small maximum degree. In 1979, Burstein proved that every graph with maximum degree 4 admits a proper acyclic coloring using 5 colors [Bur79]. We give a simple proof that (a) every graph with maximum degree 6 = 3 is acyclically 4-choosable and we prove that (b) every graph with maximum degree Delta = 4 is acyclically 5-choosable. The proof of (b) uses a backtracking greedy algorithm and Burstein's theorem.