# Planar graphs have bounded nonrepetitive chromatic number

## TL;DR

This paper proves that planar graphs and certain other graph classes can be nonrepetitively coloured with a bounded number of colours, confirming a longstanding conjecture and extending to broader graph families.

## Contribution

It establishes that planar graphs and related classes have bounded nonrepetitive chromatic number, solving a conjecture and generalising to graphs of bounded genus and excluded minors.

## Key findings

- Planar graphs have bounded nonrepetitive chromatic number.
- The result extends to graphs of bounded Euler genus.
- Applicable to graphs excluding fixed minors or topological minors.

## Abstract

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Haluszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.