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arXiv · 1202.1569

Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours

Vida Dujmović·Fabrizio Frati·Gwenaël Joret·David R. Wood
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Abstract

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound.

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Vida Dujmović, Fabrizio Frati, Gwenaël Joret, David R. Wood. 2012-02-08. Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours. https://doi.org/10.37236/3153

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