A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively k-choosable if given lists of at least k colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
Δ is cΔ2-choosable, for some constant c. We prove this result
with c=1 (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth k is nonrepetitively O(k2)-colourable.Comment: v4: Minor changes made following helpful comments by the referee