A new upper bound on the acyclic chromatic indices of planar graphs

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. It was conjectured that a′(G)≤Δ+2a′(G)≤Δ+2 for any simple graph GG with maximum degree ΔΔ. In this paper, we prove that if GG is a planar graph, then a′(G)≤Δ+7a′(G)≤Δ+7. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463–478], which says that every planar graph GG satisfies a′(G)≤Δ+12a′(G)≤Δ+12.