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Perfect matchings in random subgraphs of regular bipartite graphs
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2020-11-23 , DOI: 10.1002/jgt.22650
Roman Glebov 1 , Zur Luria 2 , Michael Simkin 3
Affiliation  

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。
更新日期:2020-11-23
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