Nonrepetitively 3-Colorable Subdivisions of Graphs with a Logarithmic Number of Subdivisions per edge

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