A bipartite graph consists of two disjoint vertex sets, where vertices of one set can only be joined with an edge to vertices in the opposite set. Hall's theorem gives a necessary and sufficient condition for a bipartite graph to have a saturating matching, meaning every vertex in one set is matched to some vertex in the other in a one-to-one correspondence. When we imagine vertices as agents and let them have preferences over other vertices, we have the classic stable marriage problem introduced by Gale and Shapley, who showed that one can always find a matching that is stable with respect to agent's preferences. These two results often clash: saturating matchings are not always stable, and stable matchings are not always saturating. I prove a simple necessary and sufficient condition for every stable matching being saturating for one side. I show that this result subsumes and generalizes some previous theorems in the matching literature. I find a necessary and sufficient condition for stable matchings being saturating on both sides, also known as perfect matchings. These results could have important implications for the analysis of numerous real-world matching markets.

中文翻译：

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饱和稳定匹配

二部图由两个不相交的顶点集组成，其中一个集的顶点只能通过一条边连接到相反集的顶点。霍尔定理给出了二部图具有饱和匹配的充分必要条件，这意味着一组中的每个顶点都与另一组中的某个顶点一一对应。当我们将顶点想象为代理并让它们对其他顶点有偏好时，我们就有了 Gale 和 Shapley 引入的经典稳定婚姻问题，他们表明，人们总是可以找到一个关于代理偏好的稳定匹配。这两个结果经常发生冲突：饱和匹配并不总是稳定的，稳定匹配并不总是饱和的。我证明了每个稳定匹配都在一侧饱和的简单充要条件。我表明这个结果包含并概括了匹配文献中的一些先前的定理。我发现稳定匹配在两边都饱和的充分必要条件，也称为完美匹配。这些结果可能对分析众多现实世界的匹配市场具有重要意义。