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

03/15/2021

∙

by   Farzam Ebrahimnejad, et al.

∙

0

∙

share

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.