Trees and Matchings

TL;DR

This paper extends Temperley's bijection to relate weighted spanning trees and perfect matchings in general planar graphs, enabling new computational and probabilistic insights into these combinatorial structures.

Contribution

It generalizes Temperley's bijection to weighted, directed planar graphs, linking spanning trees with perfect matchings in broader settings.

Findings

Establishes a one-to-one correspondence between weighted spanning trees and perfect matchings in planar graphs.

Enables computation of measures for cylinder events in random spanning trees.

Facilitates efficient sampling of perfect matchings via Wilson's algorithm in certain models.

Abstract

In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the ``square-octagon'' lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon, our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.