# Tropically planar graphs

## TL;DR

This paper investigates the properties and enumeration of tropically planar graphs, which are associated with smooth tropical plane curves, providing bounds and asymptotic results on their prevalence among planar graphs.

## Contribution

It develops necessary conditions for a graph to be tropically planar and computes their count up to genus 7, establishing bounds and asymptotic rarity.

## Key findings

- Number of tropically planar graphs computed up to genus 7
- Asymptotically 0% of connected trivalent planar graphs are tropically planar
- Provides bounds on the number of such graphs

## Abstract

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus $7$. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically $0\%$ of connected trivalent planar graphs are tropically planar.