Let $B$ be a Bipartite graph with vertex classes $V$ and $W$. There is a Complete matching from V to W if and only if for any subset S of V. The size of the neighbourhood of S $\Gamma(S)$:

$|\Gamma(S)| \geq |S|$

The "only if" part is clear because a complete matching is like an Injective function. The nontrivial part is that the condition is also necessary.