Geometry of tropical moduli spaces and linkage of graphs

TL;DR

This paper proves a linkage theorem for p-regular graphs of the same genus, showing they can be transformed into each other via edge contractions while preserving connectivity, and applies this to demonstrate connectedness of tropical moduli spaces.

Contribution

The paper introduces a linkage theorem for p-regular graphs and uses it to establish the connectedness of certain tropical moduli spaces of curves.

Findings

Graphs of the same genus are connected through edge contractions.

Tropical moduli spaces are connected through codimension one.

Connectivity is preserved under the linkage transformations.

Abstract

We prove the following "linkage" theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.