Nonrepetitive colouring via entropy compression

Vida Dujmovic, Gwenaël Joret, Jakub Kozik, David Wood

Research output: Contribution to journalArticleResearchpeer-review

22 Citations (Scopus)

Abstract

A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively ℓ-choosable if given lists of at least ℓ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that, for some constant c, every graph with maximum degree Δis cΔ2-choosable. 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 graphs with pathwidth θ are nonrepetitively O(θ2)-colourable.

Original languageEnglish
Pages (from-to)661-686
Number of pages26
JournalCombinatorica
Volume36
Issue number6
DOIs
Publication statusPublished - 2016

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