Tropical Embeddings of Metric Graphs

TL;DR

This paper demonstrates that metric graphs can be embedded as tropical curves in the plane with a number of crossings equal to their classical crossing number, linking graph theory and tropical geometry with applications in algebraic geometry.

Contribution

It establishes a method to realize metric graphs as tropical curves with minimal crossings, bridging classical graph embeddings and tropical geometry.

Findings

Metric graphs can be realized as tropical curves with minimal crossings.

The realization preserves the crossing number of the original graph.

Application to constructing rational maps with almost faithful tropicalizations.

Abstract

Every graph $\Gamma$ can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number $\text{cross}\left(\Gamma\right)$. In this paper, we prove that if $\Gamma$ is a metric graph it can be realized as a tropical curve in the plane with exactly $\text{cross}\left(\Gamma\right)$ crossings, where the tropical curve is equipped with the lattice length metric. Our result has an application in algebraic geometry, as it enables us to construct a rational map of non-Archimedean curves into the projective plane, whose tropicalization is almost faithful when restricted to their skeleton.