# Counting perfect matchings and the switch chain

## TL;DR

This paper investigates the switch Markov chain for counting perfect matchings in hereditary graph classes, identifying classes where it is ergodic, rapidly mixing, or has exponential mixing time, and provides exact algorithms for specific classes.

## Contribution

It characterizes the largest hereditary class with ergodic switch chain, introduces a new class with rapid mixing, and analyzes ergodicity and mixing times in various graph classes.

## Key findings

- Largest hereditary class with ergodic switch chain identified
- New hereditary class where the chain is rapidly mixing
- Exponential mixing time shown for a broader class

## Abstract

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in a arbitrary graph. Finally, we give exact counting algorithms for three classes.