Notes on nonrepetitive graph colouring

Abstract

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 k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring 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 Δ has a f(Δ)-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree Δ has a O(kΔ)-colouring that is nonrepetitive on paths, and a O(kΔ3)-colouring that is nonrepetitive on walks.