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Hall's marriage theorem
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{{Short description|Result in combinatorics and graph theory}} In [[mathematics]], '''Hall's marriage theorem''', proved by {{harvs|txt|first=Philip|last=Hall|authorlink=Philip Hall|year=1935}}, is a theorem with two equivalent formulations. In each case, the theorem gives a [[necessity and sufficiency|necessary and sufficient]] condition for an object to exist: * The [[Combinatorics|combinatorial]] formulation answers whether a [[Finite set|finite]] collection of [[Set (mathematics)|sets]] has a [[transversal (combinatorics)|transversal]]—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. * The [[Graph theory|graph theoretic]] formulation answers whether a finite [[bipartite graph]] has a [[perfect matching]]—that is, a way to match each vertex from one group uniquely to an adjacent vertex from the other group. Hall's condition is that any subset of vertices from one group has a [[neighbourhood (graph theory)|neighbourhood]] of equal or greater size.
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