Acyclic edge coloring of planar graphs without adjacent cycles

A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G. The acyclic edge chromatic number of G, denoted byχ′ _a (G), is the smallest number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ. In this paper, we show that χ′ _a (G) ? Δ + 2, if G has no adjacent i- and j-cycles for any i, j ∈ {3, 4, 5}, which implies a result of Hou, Liu and Wu (2012); and χ′ _a (G) ? Δ + 3, if G has no adjacent i- and j-cycles for any i, j ∈ {3, 4, 6}.