TITLE: Complete matching in bipartite graph
QUESTION [7 upvotes]: I have the following problem:
Let $G=G_2(m,n)$ be a bipartite graph with vertex classes $V_1$ and $V_2$ containing a complete matching from $V_1$ to $V_2$. Prove that there is a vertex $x\in V_1$ such that for every edge $xy\in E_G$ there is a matching from $V_1$ to $V_2$ that contains $xy$.
I tried using Hall's theorem, but worked out nothing. Maybe it is the correct way to proceed, but do not know how.
Any hints?

REPLY [2 votes]: How could the conclusion possibly fail? By Hall's theorem on the subgraph with the vertices $x,y$ removed, there would have to be a subset $U_1\subseteq V_1$ not containing $x$ such that $|\Gamma(U_1)\setminus \{y\}|<|U_1|.$ This means that in the original graph $G,$ the set $U_1$ is a non-empty proper subset of $V_1$ that is "critical" in the following sense.
A set $U_1\subseteq V_1$ is critical if
$$|U_1|=|\Gamma(U_1)|$$
where $\Gamma(U_1)$ means the neighbours of $V_1,$ as a subset of $V_2.$
Critical sets are the structure that Hall's marriage theorem gives you. If there is a non-empty proper critical subset, use induction to reduce to the case where only $\emptyset$ and $V_1$ are critical. (Alternatively, consider a minimal critical set.)