Generalised acyclic edge colourings of graphs with large girth

The r-acyclic edge chromatic number of a graph G is the minimum number of colours required to colour the edges of G in such a way that adjacent edges receive different colours and every cycle C   receives at least min{|C|,r}min{|C|,r} colours. We prove that for any integer r⩾4r⩾4, the r-acyclic edge chromatic number of any graph G   with maximum degree ΔΔ and with girth at least 3(r-1)Δ3(r-1)Δ is at most 6(r-1)Δ6(r-1)Δ.