Hall's condition for a finite bipartite graph 
with parts 
and 
is that 
for every subset 
of 
, where 
is the set of vertices
in 
adjacent to at least one vertex in 
. By Hall's theorem, this
condition is equivalent to the existence of a matching
covering every vertex in 
(Harary 1994, pp. 53-54). When 
, such a matching is a perfect matching.
Hall's Condition
See also
Hall's Theorem, Marriage Theorem, Perfect MatchingPortions of this 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:
Weisstein, Eric W., with contributions by Chris Heckman. "Hall's Condition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HallsCondition.html