Anagram-free colourings of graph subdivisions

TL;DR

This paper investigates anagram-free vertex colourings of graph subdivisions, establishing bounds on the number of colours needed and the number of division vertices per edge for various classes of graphs.

Contribution

It introduces the study of anagram-free colourings of graph subdivisions and provides bounds for general graphs, trees, complete graphs, and bounded degree trees.

Findings

Every graph has an anagram-free 8-colourable subdivision.

Constructed 10-colourable subdivisions for trees with fewer division vertices.

Established lower bounds on the number of division vertices per edge for certain graph classes.

Abstract

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex colouring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph colouring was independently introduced by Kam\v{c}ev, {\L}uczak and Sudakov and ourselves. In this paper we introduce the study of anagram-free colourings of graph subdivisions. We show that every graph has an anagram-free $8$-colourable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free $10$-colourable subdivisions with fewer division vertices per edge. Conversely, we prove lower bounds, in terms of division vertices per edge, on the anagram-free chromatic number for subdivisions of the complete graph and subdivisions of complete trees of bounded degree.