超图中完美匹配的计数阈值
Counting thresholds for perfect matchings in hypergraphs
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中文总结 AI 辅助
本文针对超图完美匹配，引入计数与近似计数阈值，证明其良定义性、非平凡性与渐近关系，并通过归约小参数情况得到改进上界，补充了超图完美匹配计数的相关理论。
中文摘要 AI 辅助
在k-均匀超图中，对于0≤d≤k-1，最小d-度是指包含任意给定d个顶点子集的边的最小数量。经典Dirac定理的一个推广保证，当顶点数为n且k整除n的k-均匀超图的最小d-度大于某个Dirac阈值时，该超图至少包含一个完美匹配。此外，Kwan、Safavi和Wang早已证明，对于d≥k/2，这类超图不仅包含一个完美匹配，还包含“大量”完美匹配，其数量至少与具有相同边密度的随机超图中的预期数量相当。不过，人们也知道这类结果无法普遍成立，因为当(d,k)=(1,3)时该结论就不成立。本文引入了“计数阈值”和“近似计数阈值”的新概念，超图在达到或超过这些阈值时，保证拥有至少对应数量的完美匹配。我们证明这些阈值对所有d、k、n都是良定义且非平凡的，它们之间存在渐近关系，最终通过归约到更小的d和k的情况，得到了改进的上界。
英文摘要
In a $k$-uniform hypergraph, the minimum $d$-degree for some $0\le d\le k-1$ is the minimum number of edges containing any given $d$-set of vertices. An extension of the classical Dirac theorem guarantees that whenever the minimum $d$-degree of a $k$-uniform $n$-vertex hypergraph, $k\mid n$, is larger than a certain Dirac threshold, it contains at least one perfect matching. Moreover, it has been known for some time, due to Kwan, Safavi, and Wang, that for $d\ge k/2$ such hypergraphs contain not only one, but ``many'' perfect matchings, that is, at least as many as are expected in a random hypergraph with the same edge density. However, it has also been known that such a result could not be hoped for in general, as it already fails for $(d,k)=(1,3)$. In this paper we introduce new notions of the \emph{counting thresholds} and \emph{approximate counting thresholds}, above which a hypergraph is guaranteed to have at least this many perfect matchings. We show that these thresholds are well-defined and nontrivial for all $d,k,n$, that they are asymptotically related, and finally, we derive improved upper bounds by reducing to cases with smaller $d$ and $k$.
