Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2-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) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a′(G) ≤ 16∆(G) for any graph G (see [2],[10]). We prove that there exists a constant c such that a′(G) ≤ ∆(G) + 2 for any graph G whose girth is at least c∆(G) log ∆(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ ∆ + 2 for almost all ∆-regular graphs.