Nonrepetitive colorings of graphs of bounded tree-width

A sequence of the form s1s2…sms1s2…sms1s2…sms1s2…sm is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [Thue type problems for graphs, points and numbers, manuscript] by proving that every outerplanar graph has a nonrepetitive 12-coloring. We also show that graphs of tree-width t   have nonrepetitive 4t4t-colorings.