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For questions about matchings in graph theory. A matching on a graph is a set of edges such that no two edges share a common vertex.

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Hall theorem with non-saturating matching

Hall's marriage theorem states that given a bipartite graph $G=(X+Y,E)$, if there is no $X$-saturating matching, there there exists $W\subseteq X$ such that $|W|>|N_G(W)|$. Is the following generaliz …