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

TL;DR

This paper demonstrates that in regular strong expander non-bipartite graphs, the ratio of near perfect to perfect matchings is polynomial, enabling efficient sampling of perfect matchings and confirming a longstanding conjecture for this graph family.

Contribution

It establishes polynomial bounds on the ratio of near perfect to perfect matchings and proves the Lovasz-Plummer conjecture for regular strong expander non-bipartite graphs.

Findings

Markov chain mixes in polynomial time for these graphs.

Number of perfect matchings is at least exponential in degree.

Confirms Lovasz-Plummer conjecture for this class of graphs.

Abstract

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.