Anagram-free Graph Colouring

TL;DR

This paper investigates anagram-free graph colouring, providing bounds on the chromatic number, constructing graphs with unbounded chromatic number despite bounded degree, and exploring extensions to trees, edges, and k-anagram-free colourings.

Contribution

It answers an open question by showing unbounded anagram-free chromatic number for graphs with maximum degree 3 and establishes bounds for trees based on radius and pathwidth.

Findings

Graphs with degree 3 can have unbounded anagram-free chromatic number.

Bounds on anagram-free chromatic number depend on tree radius and pathwidth.

Extensions to edge and k-anagram-free colourings are explored.

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.