Problem Detail: Am I correct in my observation that the cardinality of the maximum matching $M$ of a bipartite graph $G(U, V, E)$ is always equal to $min(|U|, |V|)$?
Asked By : ultrajohn
Answered By : Nicholas Mancuso
Given a bipartite graph $G = (U,V,E)$ and a maximum matching $M$ of $G$, via Konig’s Theorem we see that $|M| = |C|$ where $C$ is a minimum vertex cover for $G$. Your statement is merely an upper bound on the size of the possible matching, not a strict equality. The image on the wikipedia page provides a nice counterexample to your claim. We see that $|M| = 6$, while $min(|U|,|V|) = 7$. 
However, in the case of a complete bipartite graph $K_{n,m}$ your statement holds.
However, in the case of a complete bipartite graph $K_{n,m}$ your statement holds. Best Answer from StackOverflow
Question Source : http://cs.stackexchange.com/questions/4675
