Anagram-free colourings of graph subdivisions

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 Kamčev, Łuczak 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.