Full support of the Kasteleyn operator associated with a bipartite toroidal graph

Abstract

A perfect matching in a bipartite graph embedded on a torus defines a height function on the graph's faces and an associated height change vector in $\Z^2$. These matchings are enumerated by a combination of four evaluations of a bivariate Laurent polynomial, called Kasteleyn operator, whose coefficient of bidegree (i,j) is, up to the sign, the number of perfect matchings with height change (i,j). Therefore the Newton polygon of the Kasteleyn operator is the convex hull of the height change vectors. In this article, we prove that any point with integer coordinates in that polygon is realized by a perfect matching.