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 $\mathrm{\mid }X\mathrm{\mid }=n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]{\prod }_{i=1}^{n-1}(r-i)$ when $n<r$. When every vertex has at least two neighbors and $\mathrm{\mid }Y\mathrm{\mid }-\mathrm{\mid }X\mathrm{\mid }=t\ge 0$, the minimum is $[(n-1)t+2+b](t+1)$, where $b=\mathrm{\mid }E(G)\mathrm{\mid }-2(n+t)$. We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions.