Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours

TL;DR

This paper proves that planar graphs can be nonrepetitively coloured using only logarithmic in n number of colours, significantly improving previous bounds and addressing a major open problem in graph theory.

Contribution

The authors establish an $O( ext{log } n)$ upper bound on the nonrepetitive chromatic number of planar graphs, advancing understanding of graph colourings.

Findings

Proved $O( ext{log } n)$ upper bound for planar graphs

Improved from previous $O( ext{sqrt } n)$ bound

Addresses a key open problem in the field

Abstract

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound.