Anagram-Free Colourings of Graphs

A sequence S is called anagram-free if it contains no consecutive symbols r 1 r 2 . . . r k r k +1 . . . r 2 k such that r k +1 . . . r 2 k is a permutation of the block r 1 r 2 . . . r k . Answering a question of Erdős and Brown, Keranen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Haluszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graph G is called anagram-free if the sequence of colours on any path in G is anagram-free. We call the minimal number of colours needed for such a colouring the anagram-chromatic number of G . 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.

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