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Notes on nonrepetitive graph colouring
 Electron. J. Combin
, 2008
"... A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a sub ..."
Abstract

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A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree ∆ has a f(∆)colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree ∆ has a O(k∆)colouring that is nonrepetitive on paths, and a O(k ∆ 3)colouring that is nonrepetitive on walks. 1