Tropical Embeddings of Metric Graphs
Adan Medrano Martin del Campo, Sylvain Carpentier

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 can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number . In this paper, we prove that if is a metric graph it can be realized as a tropical curve in the plane with exactly 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsPolynomial and algebraic computation · graph theory and CDMA systems · Commutative Algebra and Its Applications
