Nonrepetitively 3-colorable subdivisions of graphs with a logarithmic number of subdivisions per edge

We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $O(\log \mathrm{\mid }V(G)\mathrm{\mid })$, $H$ is nonrepetitively 3-colorable. In fact, we show that $O(\log {\pi }^{\mathrm{\prime }}(G))$ subdivisions per edge are enough, where ${\pi }^{\mathrm{\prime }}(G)$ is the nonrepetitive chromatic index of $G$. This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable division with a linear number of subdivision vertices per edge.