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arXiv · 2301.10131

Random perfect matchings in regular graphs

Bertille Granet·Felix Joos
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Abstract

We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.

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Bertille Granet, Felix Joos. 2023-01-24. Random perfect matchings in regular graphs. https://doi.org/10.1002/rsa.21172

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