Nonrepetitive graph colouring

TL;DR

This paper surveys the field of nonrepetitive graph colourings, discussing key results, proof techniques, and open problems related to avoiding repetitive colour sequences on paths within graphs.

Contribution

It provides a comprehensive overview of nonrepetitive colourings of graphs, unifying major results and highlighting open research questions.

Findings

Thue's theorem states all paths are nonrepetitively 3-colourable.

The survey covers various graph classes and their nonrepetitive chromatic numbers.

It identifies key proof methods and open problems in the field.

Abstract

A vertex colouring of a graph $G$ is "nonrepetitive" if $G$ contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems.