# Embedding of metric graphs on hyperbolic surfaces

## TL;DR

This paper studies how metric graphs can be embedded on hyperbolic surfaces, introducing the concepts of essential genus and minimal embeddings, and providing formulas and methods for such embeddings.

## Contribution

It establishes a formula for the essential genus of metric graphs and provides explicit methods for their essential and minimal embeddings on hyperbolic surfaces.

## Key findings

- A formula to compute the essential genus of a metric graph.
- Every metric graph can be embedded on a hyperbolic surface of genus at least its essential genus.
- Explicit methods for constructing essential and minimal embeddings.

## Abstract

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ of $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.