Acyclic edge-colouring of planar graphs

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 ∆(G) is large enough then χ′a(G) = ∆(G). We settle this conjecture for planar graphs with girth at least 5 and outerplanar graphs. We also show that χ′a(G) ≤ ∆(G)+25 for all planar G, which improves a previous result by Fiedorowicz et al. [11]. 1