The tropical crossing number of a finite graph
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913998146895872 |
|---|---|
| author | Cape, Noah Morrison, Ralph |
| author_facet | Cape, Noah Morrison, Ralph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The tropical crossing number of a finite graph Cape, Noah Morrison, Ralph Combinatorics Algebraic Geometry 14T05, 05C10, 52C05 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. |
| title | The tropical crossing number of a finite graph |
| topic | Combinatorics Algebraic Geometry 14T05, 05C10, 52C05 |
| url | https://arxiv.org/abs/2508.14182 |