Acyclic edge-colouring of planar graphs

International audienceA proper edge-coloring with the property that every cycle contains edges of at least three distinct colors is called an acyclic edge-coloring. The acyclic chromatic index of a graph G, denoted χa′(G), is the minimum k such that G admits an acyclic edge-coloring with k colors. We conjecture that if G is planar and Δ(G) is large enough, then χa′(G) = Δ(G). We settle this conjecture for planar graphs with girth at least 5. We also show that χa′(G) ≤ Δ(G)+12 for all planar G, which improves a previous result by Fiedorowicz, Haluszczak, and Narayan [Inform. Process. Lett., 108 (2008), pp. 412-417]