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

TL;DR

This paper proves that every integer point within the Newton polygon of the Kasteleyn operator for bipartite toroidal graphs corresponds to a perfect matching, establishing full support of the operator.

Contribution

It demonstrates that all lattice points in the Newton polygon are realized by perfect matchings, extending understanding of the Kasteleyn operator's support.

Findings

Every integer point in the Newton polygon is realized by a perfect matching.

The Newton polygon is the convex hull of height change vectors.

The Kasteleyn operator's support covers the entire integer lattice within its polygon.

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.