Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs
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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 Ω(d)^n any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.
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