Anagram-Free Colourings of Graphs

Abstract

A sequenceSis calledanagram-freeif it contains no consecutive symbolsr1r2. . .rkrk+1. . .r2ksuch thatrk+1. . .r2kis a permutation of the blockr1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graphGis calledanagram-freeif the sequence of colours on any path inGis anagram-free. We call the minimal number of colours needed for such a colouring theanagram-chromaticnumber ofG.In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.