The Expected Number of Perfect Matchings in Cubic Planar Graphs

A well-known conjecture by Lovász and Plummer from the 1970s asserting that a bridgeless cubic graph has exponentially many perfect matchings was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture for 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. (Supported by the Ministerio de Economía y Competitividad grant MTM2017-82166-P, and by the Special Research Program F50 Algorithmic and Enumerative Combinatorics of the Austrian Science Fund.).