About acyclic edge colourings of planar graphs

Let G = (V,E) be any finite simple 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 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 χa(G). In 1991 Alon et al. in [1] proved that χa(G) ≤ 64∆(G). This bound was later improved to 16∆(G) by Molloy and Reed in [5]. In this paper we show that the acyclic chromatic index of a planar graph G without cycles of length three is at most ∆(G) + 6 and that the same holds if G is a 2 fold graph. We also prove that for planar graphs and 3-fold graphs, acyclic chromatic index does not exceed 2∆(G) + 29.