[Solved]: Size of Maximum Matching in Bipartite Graph

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.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/4675