Every k-tree and partial k-tree admits an online nonrepetitive coloring with 4^k colors, and online paths need only 12 colors.
Grytczuk, Nonrepetitive Colorings of Graphs - A Survey, Int
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A note about online nonrepetitive coloring $k$-trees
Every k-tree and partial k-tree admits an online nonrepetitive coloring with 4^k colors, and online paths need only 12 colors.