Rainbow subdivisions of cliques

TL;DR

This paper proves that sufficiently dense properly edge-coloured graphs contain rainbow subdivisions of complete graphs, using a novel approach linking random walk mixing times and graph expansion.

Contribution

It establishes a near-optimal density condition for rainbow subdivisions of cliques in properly edge-coloured graphs, connecting expansion properties with rainbow substructures.

Findings

Properly edge-coloured graphs with $n (\log n)^{53}$ edges contain rainbow $K_m$ subdivisions.

The result is sharp up to a polylogarithmic factor.

The proof uses the relationship between random walk mixing times and graph expansion.

Abstract

We show that for every integer $m \ge 2$ and large $n$, every properly edge-coloured graph on $n$ vertices with at least $n (\log n)^{53}$ edges contains a rainbow subdivision of $K_m$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.