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Abstract: An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number such that there is an acyclic edge coloring using colors and it is denoted by . From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors. [Copyright &y& Elsevier]

Abstract: An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-) cycles. The acyclic chromatic index of a graph is the minimum number such that there is an acyclic edge coloring using colors and is denoted by . Let denote the maximum degree of a vertex in a graph . A complete bipartite graph with vertices on each side is denoted by . Alon, McDiarmid and Reed observed that for every prime . In this paper we prove that when is prime. Basavaraju, Chandran and Kummini proved that when is odd, which combined with our result implies that when is an odd prime. Moreover we show that if we remove any edge from , the resulting graph is acyclically -edge-colorable. [Copyright &y& Elsevier]