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arXiv · 2404.14020

Perfect Matching in Product Graphs and in their Random Subgraphs

Sahar Diskin·Anna Geisler
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Abstract

For $t\in \mathbb{N}$ and ever $i\in [t]$, let $H_i$ be a $d_i$-regular connected graph with $1<|V(H_i)|\le M$ for some integer $M\ge 2$. Let $G=\square_{i=1}^tH_i$ be the $t$-dimensional Cartesian product of $H_1,\ldots, H_t$. We prove that if $t\ge 2\ln M$ then $G$ has a (nearly-)perfect matching. We further show that this bound on the dimension is tight up to a constant factor. Then, considering the random graph process on $G$, we generalise the result of Bollob\'as on the binary hypercube $Q^t$, showing that with high probability, the hitting times for minimum degree one, connectivity, and the existence of a (nearly-)perfect matching in the random graph process on $G$ are the same.

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Sahar Diskin, Anna Geisler. 2024-04-22. Perfect Matching in Product Graphs and in their Random Subgraphs. https://arxiv.org/abs/2404.14020

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