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 Bollobás 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.
From: Sahar Diskin [view email]
[v1]
Mon, 22 Apr 2024 09:35:25 UTC (113 KB)
[v2]
Thu, 2 Jan 2025 09:28:20 UTC (114 KB)
[v3]
Mon, 25 Aug 2025 10:11:57 UTC (102 KB)
[v4]
Tue, 1 Sep 2026 17:41:26 UTC (48 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。