The tropical crossing number of a finite graph

TL;DR

This paper introduces the tropical crossing number for finite graphs, demonstrating its range, computational methods for specific graphs, and its potential quadratic growth relative to the number of vertices.

Contribution

It defines the tropical crossing number for non-metric graphs, proves its unboundedness, and develops computational techniques to determine it for particular graphs.

Findings

Existence of graphs with any prescribed tropical crossing number

Development of computational methods for specific graphs

Tropical crossing number can grow quadratically with vertices

Abstract

In 2015, Cartwright et al. showed that any $3$-regular metric graph arises as the skeleton of a tropical plane curve with nodes allowed. They introduced the tropical crossing number of a metric graph as the minimum number of nodes required for that graph with the prescribed lengths. We introduce the tropical crossing number of a finite, non-metric graph, the minimum number of nodes required to achieve that graph with any lengths on its edges. We prove that for any positive integer $d$ there exists a graph whose tropical crossing number is equal to $d$; moreover, this graph can be chosen with any prescribed graph-theoretic crossing number at most $d$. We then introduce and use computational methods to find the tropical crossing number of the smallest non-tropically planar graph, the lollipop graph of genus $3$. We also show that our tropical crossing number can grow quadratically in the number of vertices of the graph.