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, Kardos, King, Kr\'al and Norine (2011). On the other hand, Chudnovsky and Seymour (2012) proved the conjecture in the special case of cubic planar graphs. 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 labelled bridgeless cubic planar graphs is asymptotically c^n where c > 1 is an explicit algebraic number. 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.

This is joint work with Clément Requilé and Juanjo Rué.

https://zoom.us/j/94541219182?pwd=ZDdOT1poNUxTNlBFUE93M1dIZ0Eydz09

Meeting-ID: 945 4121 9182, Kenncode: Let2Vh