Acyclic List Colouring Locally Planar Graphs

TL;DR

This paper establishes that locally planar graphs can be acyclically list-coloured with up to 9 colours, advancing understanding of colourability bounds for this class of graphs.

Contribution

It proves the first known upper bound of 9 for acyclic list colouring of locally planar graphs, improving previous bounds for similar graph classes.

Findings

Locally planar graphs are acyclically 9-list-colourable.

No prior fixed bound was known for acyclic list colouring of locally planar graphs.

The result extends the understanding of colourability in graph theory.

Abstract

A (vertex) colouring of 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\textemdash no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known.