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

An acyclic edge coloring of a graph GG is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a′(G)a′(G) of GG is the smallest integer kk such that GG has an acyclic edge coloring using kk colors. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a′(G)≤Δ+2a′(G)≤Δ+2 for any simple graph GG with maximum degree ΔΔ. In this paper, we show that if GG is a planar graph without a 3-cycle adjacent to a 4-cycle, then a′(G)≤Δ+2a′(G)≤Δ+2, i.e., this conjecture holds.