Perfect Matching in Product Graphs and in their Random Subgraphs

TL;DR

This paper proves the existence of nearly-perfect matchings in Cartesian products of regular graphs and extends results on random subgraphs, including hitting times for key properties, with new tools for percolation analysis.

Contribution

It establishes conditions for perfect matchings in product graphs and generalizes Bollobás's results on hypercubes to broader graph classes.

Findings

Nearly-perfect matchings exist in product graphs when t ≥ 5C.

Hitting times for key properties coincide in the random graph process.

Develops new tools for analyzing perfect matchings under percolation.

Abstract

For $t \in \mathbb{N}$ and every $i\in[t]$, let $H_i$ be a $d_i$-regular connected graph, with $1<|V(H_i)|\le C$ for some integer $C\ge 2$. Let $G=\square_{i=1}^tH_i$ be the Cartesian product of $H_1, \ldots, H_t$. We show that if $t\ge 5C$ then $G$ contains a (nearly-)perfect matching. 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. As a byproduct, we develop several tools which may be of independent interest in a more general setting when one seeks to establish the typical existence of a perfect matching under percolation.