Sharp lower bounds for the number of maximum matchings in bipartite multigraphs

We study the minimum number of maximum matchings in a bipartite multigraph $G$ with parts $X$ and $Y$ under various conditions, refining the well-known lower bound due to M. Hall. When $\mid X\mid =n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!\left(k-r+1\right)$ when $n\ge r$ and is $\left[r+n\left(k-r\right)\right]{\prod }_{i=1}^{n-1}\left(r-i\right)$ when $n\le r$. When every vertex has at least two neighbors and $\mid Y\mid -\mid X\mid =t\ge 0$, the minimum is $\left[\left(n-1\right)t+2+b\right]\left(t+1\right)$, where $b=\mid E\left(G\right)\mid -2\left(n+t\right)$. We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions.