TITLE: Prove that a $k$-regular bipartite graph has a perfect matching
QUESTION [11 upvotes]: Prove that a $k$-regular bipartite graph has a perfect matching by using Hall's theorem.

Let $S$ be any subset of the left side of the graph. The only thing I know is the number of things leaving the subset is $|S|\times k$.

REPLY [2 votes]: Let $S$ be any subset of vertices in the left vertex set of the $k$-regular bipartite graph.  The number of edges adjacent to vertices in $S$ is exactly $|S|~ k$. Since the number of edges incident to each vertex in the right vertex set of the bipartite graph is exactly $k$, any set of $|S|~k$ edges in the bipartite graph will be incident to $|S|$ or more vertices in the right vertex set. (For example, fewer than $|S|$ vertices in the right vertex set can `accomodate' at most $(|S|-1)k$ edges.) Thus, the number of neighbors of $S$ is at least $|S|$.  By Hall's theorem, the graph has a perfect matching.