发表机构
ETH（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文探讨超图中完美匹配的存在条件，揭示分数完美匹配与Feige概率不等式间的联系，并给出该猜想的简洁自包含证明。
AI 中文摘要
一个$n$顶点图需要多大的最小度才能确保它包含一个完美匹配？Dirac定理指出，在偶数个顶点的图中，如果每个顶点的度数至少为$n/2$，则该图具有此性质。在这篇简短的说明性笔记中，旨在用于课堂教学，我们讨论这一陈述如何推广到超图。特别是，我们强调了超图中的分数完美匹配与关于非负随机变量的概率不等式之间的一个优雅联系，该不等式由Feige猜想。我们还给出了Feige猜想的一个非常简短的独立证明。
英文摘要
How large of a minimum degree does an $n$-vertex graph need before we are sure that it contains a perfect matching? Dirac's theorem states that a graph on an even number of vertices in which each vertex has degree at least $n/2$ has this property. In this short expository note, intended to be used in the classroom, we discuss how this statement generalizes to hypergraphs. In particular, we highlight an elegant connection between fractional perfect matchings in hypergraphs and a probabilistic inequality about nonnegative random variables, which was conjectured by Feige. We also present a very short self-contained proof of Feige's conjecture.
