Counting perfect matchings of cubic graphs in the geometric dual

Lovász and Plummer conjectured, in the mid 1970’s, that every cubic graph G with no cutedge has an exponential in |V (G)| number of perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation has at least 3φ (G)|/72 distinct perfect matchings, where φ is the golden ratio. Our work builds on a novel approach relating Lovász and Plummer’s conjecture and the number of so called groundstates of the widely studied Ising model from statistical physics.