2025/08/19 by Cape, Noah, Morrison, Ralph · 1 citation
#05C10 #14T05 #52C05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.14182
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.