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Nonrepetitive choice number of trees

A nonrepetitive coloring of a path is a coloring of its vertices such that the sequence of colors along the path does not contain two identical, consecutive blocks. The remarkable construction of Thue asserts that 3 colors are enough to color nonrepetitively paths of any length. A nonrepetitive coloring of a graph is a coloring of its vertices such that all simple paths are nonrepetitively colored. Assume that each vertex $v$ of a graph $G$ has assigned a set (list) of colors $L_v$. A coloring is chosen from $\{L_v\}_{v\in V(G)}$ if the color of each $v$ belongs to $L_v$. The Thue choice number of $G$, denoted by $π_l(G)$, is the minimum $k$ such that for any list assignment $\set{L_v}$ of $G$ with each $|L_v|\geq k$ there is a nonrepetitive coloring of $G$ chosen from $\{L_v\}$. Alon et al. (2002) proved that $π_l(G)=O(Δ^2)$ for every graph $G$ with maximum degree at most $Δ$. We propose an almost linear bound in $Δ$ for trees, namely for any $\epsi>0$ there is a constant $c$ such that $π_l(T)\leq cΔ^{1+\epsi}$ for every tree $T$ with maximum degree $Δ$. The only lower bound for trees is given by a recent result of Fiorenzi et al. (2011) that for any $Δ$ there is a tree $T$ such that $π_l(T)=Ω(\frac{\logΔ}{\log\logΔ})$. We also show that if one allows repetitions in a coloring but still forbid 3 identical consecutive blocks of colors on any simple path, then a constant size of the lists allows to color any tree.