Acyclic edge colouring of plane graphs

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G), is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar graphs with girth at least 5. We also show that χa(G) ≤ ∆(G)+ 12 for all planar G, which improves a previous result by Fiedorowicz et al. [12].