Problem 4 (20 points)
Consider a bipartite graph \( G=(V, E) \) with parts \( A, B \) such that \( |A|=|B| \).
Show that if \( \left|N_{G}(S)\right|>|S| \) for all \( S \) such that \( S \subsetneq A(S \) is a proper subset of \( A) \) and \( S \neq \emptyset \), then for any edge \( e \in E, G \) contains a perfect matching that includes \( e \). Note that \( N_{G}(S):=\{u \in V: u \) is adjacent in graph \( G \) to a vertex in \( S\} \) is the set of neighbors in graph \( G \) of all vertices in \( S \subseteq V \).
