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 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 , denoted by χ a ′ ( G ) . 1, Alon et al. conjectured that for any graph G it holds that χ a ′ ( G ) ≤ Δ ( G ) + 2 ; here Δ ( G ) stands for the maximum degree of G . s 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 χ a ′ ( G ) ≤ Δ ( 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 χ a ′ ( G ) ≤ Δ ( G ) + 15 .