Acyclic edge colourings of graphs with large girth
  An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)\Delta$ colours. 
Xing Shi Cai, Guillem Perarnau, Bruce Reed, Adam Bene Watts
math.CO cs.DM