A proper coloring of the edges of a graph G is called acyclic if there is no two‐colored cycle in G . The acyclic edge chromatic number of G , denoted by a'(G) , is the least number of colors in an acyclic edge coloring of G . For certain graphs G , a'(G)\geq Δ(G)+2 where Δ(G) is the maximum degree in G . It is known that a'(G)≤ Δ + 2 for almost all Δ ‐regular graphs, including all Δ ‐regular graphs whose girth is at least cΔ log Δ . We prove that determining the acyclic edge chromatic number of an arbitrary graph is an NP‐complete problem. For graphs G with sufficiently large girth in terms of Δ(G) , we present deterministic polynomial‐time algorithms that color the edges of G acyclically using at most Δ(G)+2 colors.
