  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.
