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.



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> If $\delta(G) \geq n/2$, how many perfect matchings must $G$ contain?
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For $k$-regular bipartite graphs the answer to the corresponding question is due to [Schrijver](http://homepages.cwi.nl/~lex/files/countpms2.pdf). 


A [closely related question](https://mathoverflow.net/questions/237976/on-schrijvers-lower-bound-for-the-number-of-perfect-matchings) did not focus on the range of degrees where perfect matchings are guaranteed·



Number of perfect matchings in bipartite graph with given minimum degree
It's [a theorem of Marshall Hall](http://www.ams.org/journals/bull/1948-54-10/S0002-9904-1948-09098-X/) that in a bipartite graph of minimum degree $r$, where there exists a perfect matching, there will be at least $r!$ perfect matchings. In your case the conditional disappears and you get a lower bound of $\lceil\frac{n}{2}\rceil!$ perfect matchings. 


This lower bound is tight, and it's achieved by the graph on vertices $u\_1,v\_1,\dots,u\_n,v\_n$, where there is an edge betweenn $u\_i$ and $v\_j$ for all $i,j\le \lceil\frac{n}{2}\rceil$, and an edge for all $i\le j$. A perfect matching in this graph must contain edges $u\_iv\_i$ for all $i > \lceil \frac{n}{2}\rceil$, therefore it contains exactly $\lceil\frac{n}{2}\rceil!$ perfect matchings.


