%0 Journal Article
%T Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours
%A Vida DujmoviŁż
%A Fabrizio Frati
%A GwenaŁżl Joret
%A David R. Wood
%J Computer Science
%D 2012
%I arXiv
%X 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.
%U http://arxiv.org/abs/1202.1569v2