Acyclic edge-colouring of planar graphs. Extended abstract

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 χ a ′ ( G ) is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and Δ ( G ) is large enough then χ a ′ ( G ) = Δ ( G ) . We settle this conjecture for planar graphs with girth at least 5 and outerplanar graphs. We also show that if G is planar then χ a ′ ( G ) ⩽ Δ ( G ) + 25 .