  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.
