Editing Hall's marriage theorem (section)

== Fractional matching variant ==
A ''fractional matching'' in a graph is an assignment of non-negative weights to each edge, such that the sum of weights adjacent to each vertex is at most 1. A fractional matching is ''X''-perfect if the sum of weights adjacent to each vertex is exactly 1. The following are equivalent for a bipartite graph ''G'' = (''X+Y, E''):<ref>{{Cite web|title=co.combinatorics - Fractional Matching version of Hall's Marriage theorem|url=https://mathoverflow.net/questions/271939/fractional-matching-version-of-halls-marriage-theorem|access-date=2020-06-29|website=MathOverflow}}</ref>

* ''G'' admits an X-perfect matching.
* ''G'' admits an X-perfect fractional matching. The implication follows directly from the fact that ''X''-perfect matching is a special case of an ''X''-perfect fractional matching, in which each weight is either 1 (if the edge is in the matching) or 0 (if it is not).
* ''G'' satisfies Hall's marriage condition. The implication holds because, for each subset ''W'' of ''X'', the sum of weights near vertices of ''W'' is |''W''|, so the edges adjacent to them are necessarily adjacent to at least ''|W|'' vertices of ''Y''.