Geometric Bijections Between Spanning Trees and Break Divisors

TL;DR

This paper explores geometric bijections between spanning trees and break divisors in graphs, introducing edge ordering maps and proving their relation to Bernardi bijections, with implications for combinatorial algorithms.

Contribution

It provides a new geometric framework for constructing and understanding bijections between spanning trees and break divisors, including the class of edge ordering maps.

Findings

Introduction of edge ordering maps with good algorithmic properties

Proof that planar Bernardi bijections are geometric

Sharpened results on Bernardi torsors

Abstract

The Jacobian group ${\rm Jac}(G)$ of a finite graph $G$ is a group whose cardinality is the number of spanning trees of $G$. $G$ also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et al. obtained a polyhedral decomposition of the tropical Jacobian where vertices and cells correspond to elements of ${\rm Jac}(G)$ and spanning trees of $G$, respectively. We give a combinatorial description of bijections coming from this geometric setting. This provides a new geometric method for constructing bijections in combinatorics. We introduce a special class of geometric bijections that we call edge ordering maps, which have good algorithmic properties. Finally, we study the connection between our geometric bijections and the class of bijections introduced by Bernardi; in particular we prove a conjecture of Baker that planar Bernardi bijections are "geometric". We also give sharpened versions of results by Baker and Wang on Bernardi torsors.