Acyclic edge coloring of planar graphs with Δ colors

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that Delta(G) + 2 colors suffice for an acyclic edge coloring of every graph G (Fiamcik, 1978 [8]). The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is Delta + 12 (Basavaraju and Chandran, 2009[3]). In this paper, we study simple planar graphs which need only Delta(G) colors for an acyclic edge coloring. We show that a planar graph with girth g and maximum degree Delta admits such acyclic edge coloring if g >= 12, or g >= 8 and Delta >= 4, or g >= 7 and Delta >= 5, or g >= 6 and Delta >= 6, org >= 5 and Delta >= 10. Our results improve some previously known bounds. (C) 2012 Elsevier B.V. All rights reserved.