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

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. It was conjectured that a' (G) <= Delta + 2 for any simple graph G with maximum degree Delta. In this paper, we prove that if G is a planar graph, then a' (G) <= Delta + 7. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Haver, T. Muller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463-478], which says that every planar graph G satisfies a'(G) <= Delta + 12. (c) 2012 Elsevier Ltd. All rights reserved.