On the expected number of perfect matchings in cubic planar graphs

TL;DR

This paper analyzes the expected number of perfect matchings in random bridgeless cubic planar graphs, establishing an asymptotic growth rate and connecting it to models in statistical physics.

Contribution

It provides the first asymptotic estimate for the expected number of perfect matchings in random bridgeless cubic planar graphs, linking combinatorics with the Ising model.

Findings

Expected number of perfect matchings grows as cγ^n with γ ≈ 1.14196

Derived explicit asymptotic formulas for labeled graphs

Provided lower bounds for unlabeled cubic planar graphs

Abstract

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.