Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs.

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $\Omega(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.