Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2021
- DOI:
- 10.48550/arXiv.2103.08683
- arXiv:
- arXiv:2103.08683
- Bibcode:
- 2021arXiv210308683E
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Mathematics - Combinatorics
- E-Print:
- 14 pages, no figures