Acyclic edge colouring of planar graphs without short cycles

Let G = (V, E) be any finite graph. A mapping C : E -> [k] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i on, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by chi(a)'(G). In 2001, Alon et al. conjectured that for any graph G it holds that chi(a)'(G) <= Delta(G) + 2; here Delta(G) stands for the maximum degree of G. In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4, 6, 8 and 9. We also show that chi(a)'(G) <= Delta(G) + 1 if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then chi(a)'(G) <= (G) + Delta(G) + 15. (C) 2009 Elsevier B.V. All rights reserved.