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.

翻译：我们显示,接近完美匹配数量与美元-经常强扩张器(非两边)图中与2美元顶点的完美匹配数量之比,是一个以美元计的多元数字,因此,Jerrum和Sinclair Markov 链条[JS89]混合在多元时间,产生一个(几乎)单一随机的完美匹配。此外,我们证明,此类图表至少有1美元/Omega(d) ⁇ n 任何完美的匹配,从而证明了这一组图表的Lovasz-Plummer预测值[LP86]。