Counting perfect matchings of cubic graphs in the geometric dual
arXiv:1010.5918
Abstract
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 3 times the golden ratio to |V(G)|/72 distinct perfect matchings. 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.
18 pages, 8 figures