Nonrepetitive colouring via entropy compression

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.

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Correspondence to David R. Wood.

Additional information

Supported by NSERC and an Endeavour Fellowship from the Australian Government.

Supported by the Belgian Fund for Scientific Research (F.R.S.-FNRS), by an Endeavour Fellowship from the Australian Government, and by a DECRA Fellowship from the Australian Research Council.

Supported by the Polish National Science Center (DEC-2011/01/D/ST1/04412).

Research supported by the Australian Research Council.

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Dujmović, V., Joret, G., Kozik, J. et al. Nonrepetitive colouring via entropy compression. Combinatorica 36, 661–686 (2016). https://doi.org/10.1007/s00493-015-3070-6

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Mathematics Subject Classification (2000)

  • 05C15