Anagram-free Graph Colouring

Abstract

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with 'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov. Minor changes based on referee feedbac