Acyclic List Colouring Locally Planar Graphs

Abstract

A (vertex) colouring of a graph is \emph{acyclic} if it contains no bicoloured cycle. In 1979, Borodin proved that planar graphs are acyclically 5-colourable. In 2010, Kawarabayashi and Mohar proved that locally planar graphs are acyclically 7-colourable. In 2002, Borodin, Fon-Der-Flaass, Kostochka, Raspaud, and Sopena proved that planar graphs are acyclically 7-list-colourable. We prove that locally planar graphs are acyclically 9-list-colourable - no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known. We further show that triangle-free locally planar graphs are acyclically 8-list-colourable.