Hall's Theorem
Deficiency of Bipartite Graph
\[ \text{let}\ G = (X + Y, E) \\ \mathrm{def}(G;X) = \max_{S\subseteq X} |S| - |\mathrm{Adj}(S)| \]
since \(S\) can be \(\emptyset\), \(\mathrm{def}(G;X)\ge 0\).
extended Hall's Theorem
Every bipartite graph \(G = (X+Y, E)\) admits a matching in which at most \(def(G;X)\) vertices of \(X\) are unmatched. (from Wikipedia).