Nonrepetitive Colouring via Entropy Compression

TL;DR

This paper improves bounds on nonrepetitive graph colourings using entropy compression, showing all graphs with maximum degree Δ are nonrepetitively Δ^2-choosable, subdivisions are 5-choosable, and graphs with pathwidth k are O(k^2)-colourable.

Contribution

It establishes tighter bounds on nonrepetitive colourings for various graph classes using entropy compression methods, including degree-based, subdivision, and pathwidth results.

Findings

Graphs with maximum degree Δ are nonrepetitively Δ^2-choosable.

Subdivisions of graphs are nonrepetitively 5-choosable.

Graphs with pathwidth k are nonrepetitively O(k^2)-colourable.

Abstract

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that every graph with maximum degree $\Delta$ is $c\Delta^2$-choosable, for some constant $c$. We prove this result with $c=1$ (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5-choosable. The proofs of both these results are based on the Moser-Tardos entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that every graph with pathwidth $k$ is nonrepetitively $O(k^{2})$-colourable.