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

#### Abstract

A vertex colouring of a graph is *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 *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.

#### Keywords

Nonrepetitive graph colouring; Planar graphs