Acyclic Chromatic Indices of Planar Graphs with Girth At Least 4

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic indexa ′ ( G )of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiam c ̌ ik (Math. Slovaca 28 (1978), 139–145) and later Alon et al. (J Graph Theory 37 (2001), 157–167) conjectured thata ′ ( G ) ≤ Δ + 2 for any simple graph G with maximum degree Δ. In this article, we confirm this conjecture for planar graphs of girth at least 4.