Let $G$ be a spanning subgraph of $K_{n,n}$ with minimum degree $\delta(G) \geq n/2$. It's easy to show using Hall's theorem that $G$ has a perfect matching, and the example of two disjoint copies of $K_{\lfloor n/2 \rfloor + 1, \lceil n/2 \rceil - 1}$ side by side shows that $n/2$ is sharp. This extremal example "almost" has lots of perfect matchings—it has lots of matchings covering almost all of the vertices.

If $\delta(G) \geq n/2$, how many perfect matchings must $G$ contain?

For $k$-regular bipartite graphs the answer to the corresponding question is due to Schrijver.

A closely related question did not focus on the range of degrees where perfect matchings are guaranteed·