Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary $k$-regular bipartite graphs $G$ on $2n$ vertices for all $k=\Omega(n)$. Surprisingly, this is not the case for smaller values of $k$. We construct sparse bipartite $k$-regular graphs in which the last isolated vertex disappears long before a perfect matching appears.

中文翻译：

---

规则二部图随机子图中的完美匹配

考虑一个随机过程，其中图 $G$ 的边以随机顺序一条一条添加。一个经典的结果表明，如果 $G$ 是完全图 $K_{2n}$ 或完全二部图 $K_{n,n}$，那么通常完美匹配出现在最后一个孤立顶点消失的时刻。我们将此结果扩展到所有 $k=\Omega(n)$ 的 $2n$ 顶点上的任意 $k$-正则二分图 $G$。令人惊讶的是，对于较小的 $k$ 值，情况并非如此。我们构建了稀疏二部 $k$-regular 图，其中最后一个孤立的顶点在完美匹配出现之前很久就消失了。