About acyclic edge colourings of planar graphs

Let G=(V,E) be any finite simple graph. A mapping 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 by all the edges which have either colour i or j 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 and is denoted by .In 1991, Alon et al. [N. Alon, C.J.H. McDiarmid, B.A. Reed, Acyclic coloring of graphs, Random Structures and Algorithms 2 (1991) 277–288] proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed in [M. Molloy, B. Reed, Further algorithmic aspects of the local lemma, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 524–529].In this paper we prove that for a planar graph G without cycles of length three and that the same holds if G has an edge-partition into two forests. We also show that if G is planar.