Let $M$ and $M'$ be two different matchings (may not be edge disjoint) in a bipartite graph $G$. If $|M'| \gt 2 |M|$ then show that there is an edge $e' \in M'$ such that $|M| \cup \{e'\}$ is a matching in G.
How do we go about proving this?
2026-03-25 01:20:32.1774401632
If $|M'| \gt 2 |M|$ then show that there is an edge $e' \in M'$ such that $|M| \cup \{e'\}$ is a matching in G.
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If it is not possible to find such an edge $e'$, that means each edge of $M'$ must touch at least one of the $2|M|$ vertices covered by $M$. Can you see how the pigeonhole principle can help here?