Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by x?? _a (G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree ?? and girth g(G), and let 1 ? r ? 2?? be an integer. In this paper, it is shown that there exists a constant c > 0 such that if $g(G) \geqslant \frac{{c\Delta }} {r}\log \left( {\Delta ^2 /r} \right)$ then x?? _a (G) ? ??+r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that x?? _a (G) = ?? if ?? ? 4 and g(G) ? 4; or ?? ? 3 and g(G) ? 5.