Given a set 
,
let 
be the set of neighbors of 
. Then the bipartite graph 
with bipartitions 
and 
has a perfect matching iff 
for all subsets 
of 
.
Hall's Condition
See also
Diversity Condition, Hall's Theorem, Marriage Theorem, Perfect MatchingThis entry contributed by Chris Heckman
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References
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Referenced on Wolfram|Alpha
Hall's ConditionCite this as:
Heckman, Chris. "Hall's Condition." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HallsCondition.html