Optimal acyclic edge-coloring of cubic graphs

An acyclic edge-coloring of a graph is a proper edge-coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge-coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K4 or K3,3; the acyclic chromatic index of K4 and K3,3 is 5. This result has previously been published by Fiamcik, but his published proof was erroneous.