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

Abstract

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. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a′(G)≤Δ+2 for any simple graph G with maximum degree Δ. In this paper, we show that if G is a planar graph without a 3-cycle adjacent to a 4-cycle, then a′(G)≤Δ+2, i.e., this conjecture holds.