Rainbow subdivisions of cliques
Tao Jiang, Shoham Letzter, Abhishek Methuku, Liana Yepremyan

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 and large , every properly edge-coloured graph on vertices with at least edges contains a rainbow subdivision of . 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.
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Limits and Structures in Graph Theory · Stochastic processes and statistical mechanics
