Kasteleyn cokernels and perfect matchings on planar bipartite graphs

TL;DR

This paper explores the relationship between Kasteleyn cokernels and perfect matchings in planar bipartite graphs, establishing a canonical group action that generalizes known bijections and deepens understanding of graph invariants.

Contribution

It introduces a canonical simply transitive group action of the Kasteleyn cokernel on perfect matchings for a class of planar bipartite graphs, extending Bernardi's bijection results.

Findings

Established a group action of the Kasteleyn cokernel on perfect matchings.

Extended Bernardi's bijection to a broader class of graphs.

Provided new insights into the structure of perfect matchings and graph invariants.

Abstract

The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning trees of a planar bipartite graph and elements of its Jacobian. In this paper, we explore an analogue of Bernardi's results, providing a canonical simply transitive group action of the Kasteleyn cokernel of a planar bipartite graph on its set of perfect matchings, when the planar bipartite graph in question is of the form $G^+$, as defined by Kenyon, Propp and Wilson.