Thue choosability of trees

International audienceA vertex colouring of a graph G is nonrepetitive if for any path $P = (v_1, v_2,\dots., v_{2r})$ in $G$, the first half is coloured differently from the second half. The Thue choice number of $G$ is the least integer $l$ such that for every $l$-list assignment $L$ of $G$, there exists a nonrepetitive $L$-colouring of $G$. We prove that for any positive integer $l$, there is a tree $T$ with $\pi_{ch}(T) > l$. On the other hand, it is proved that if $G'$ is a graph of maximum degree $\Delta$, and $G$ is obtained from $G'$ by attaching to each vertex $v$ of $G'$ a graph of tree-depth at most $d$ rooted at $v$, then $\pi_{ch}(G)\leq c(\Dela, d)$ for some constant $c(\Delta, d)$ depending only on $\Delta$ and $d$