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

TL;DR

This paper proves that subdividing each edge of any graph logarithmically many times ensures the resulting graph is nonrepetitively 3-colorable, improving previous linear bounds and answering an open question.

Contribution

It establishes that logarithmic subdivisions per edge suffice for nonrepetitive 3-colorability, advancing understanding of graph colorings after subdivisions.

Findings

Logarithmic subdivisions guarantee nonrepetitive 3-colorability.

Improves previous linear bounds on subdivisions needed.

Answers an open question by Wood about subdivision bounds.

Abstract

We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $O(\log |V(G)|)$, $H$ is nonrepetitively 3-colorable. In fact, we show that $O(\log \pi'(G))$ subdivisions per edge are enough, where $\pi'(G)$ 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.