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

  • Vida Dujmović
  • Gwenaël Joret
  • Fabrizio Frati
  • David R. Wood
Keywords: Nonrepetitive graph colouring, Planar graphs


A vertex colouring of a graph is 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 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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