Counting perfect matchings in the geometric dual

Lovász and Plummer conjectured, in the mid 1970ʼs, that every bridgeless cubic graph has exponentially many perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation (planar 3-tree) has at least 3 ϕ | V ( G ) | / 72 distinct perfect matchings, where ϕ is the golden ratio. Our work builds on a novel approach relating Lovász and Plummerʼs claim and the number of so called groundstates of the widely studied Ising model.