The acyclic edge coloring of planar graphs without a 3-cycle adjacent to a 4-cycle

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^'(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamcik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a^'(G)@[email protected]+2 for any simple graph G with maximum degree @D. In this paper, we show that if G is a planar graph without a 3-cycle adjacent to a 4-cycle, then a^'(G)@[email protected]+2, i.e., this conjecture holds.