On the Number of Perfect Matchings in a Bipartite Graph

In this paper we show that, with 11 exceptions, any matching covered bipartite graph on $n$ vertices, with minimum degree greater than two, has at least $2n-4$ perfect matchings. Using this bound, which is the best possible, and McCuaig's theorem [W. McCuaig, J. Graph Theory, 38 (2001), pp. 124--169] on brace generation, we show that any brace on $n$ vertices has at least $(n-2)^2/8$ perfect matchings. A bi-wheel on $n$ vertices has $(n-2)^2/4$ perfect matchings. We conjecture that there exists an integer $N$ such that every brace on $n\geq N$ vertices has at least $(n-2)^2/4$ perfect matchings. [PUBLICATION ABSTRACT]