Acyclic Colourings of Planar Graphs with Large Girth

Abstract

A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number χa = 5 and girth g = 4. It is proved here that a planar graph satisfies χa ⩽ 4 if g ⩾ 5 and χa ⩽ 3 if g ⩾ 7.