Ryser猜想的极值超图：家乡超图
Extremal Hypergraphs for Ryser's Conjecture: Home-Base Hypergraphs
摘要：
Ryser猜想指出任何$r$ -partite $r$ -uniform超图的顶点覆盖的大小至多是最大匹配的大小的$r - 1$倍.对于$r = 2$，这个猜想只是柯尼希定理，并且每个二分图都是它的紧性的证明。Aharoni也用拓扑方法证明了$r = 3$的猜想，但证明没有给出关于极$3$ -一致超图的信息.本文的目的是刻画那些满足Aharoni定理的紧超图。我们的证明这一特征也是基于拓扑机器，特别是利用结果的（拓扑）连通性的独立复杂的线图的链接图的$3$ -均匀Ryser极值超图，在一个单独的文件。当前的文件包含第二，结构超图理论的部分的论点，在那里我们使用的链接图的线图的信息，以钉下来的结构，我们称之为\emph{home-base hypergraph}的元素。虽然对每一个正整数$k \in \mathbb{N}$都存在一个匹配数为$k$的极小家基超图，但匹配数为$k$的家基超图远非唯一.它们有无限多个，每个都是由两种不同基本结构的$k$副本组成，它们的超边可以以各种受限但复杂的方式相交。我们的特征也证明了一个古老的和广泛的开放加强Ryser的猜想，由于Lovász，$3$ -一致极值的情况下，也就是说，超图与$τ= 2 ν$。
英文摘要：
Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply König's Theorem and every bipartite graph is a witness for its tightness. The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem. Our proof of this characterization is also based on topological machinery, particularly utilizing results on the (topological) connectedness of the independence complex of the line graph of the link graphs of $3$-uniform Ryser-extremal hypergraphs, developed in a separate paper. The current paper contains the second, structural hypergraph-theoretic part of the argument, where we use the information on the line graph of the link graphs to nail down the elements of a structure we call \emph{home-base hypergraph}. While there is a single minimal home-base hypergraph with matching number $k$ for every positive integer $k \in \mathbb{N}$, home-base hypergraphs with matching number $k$ are far from being unique. There are infinitely many of them and each of them is composed of $k$ copies of two different kinds of basic structures, whose hyperedges can intersect in various restricted, but intricate ways. Our characterization also proves an old and wide open strengthening of Ryser's Conjecture, due to Lovász, for the $3$-uniform extremal case, that is, for hypergraphs with $τ= 2 ν$.
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