Limiting crossing numbers for geodesic drawings on the sphere

TL;DR

This paper studies the properties of random geodesic drawings of bipartite graphs on the sphere, focusing on intersection graphs and their limiting behavior, revealing measure-independent edge densities but measure-dependent triangle densities.

Contribution

It introduces a model for random geodesic graph drawings on the sphere and analyzes the convergence and properties of their intersection graphs, including measure-independent edge density and measure-dependent triangle density.

Findings

Intersection graphs form convergent graph sequences.

Edge density of the limit is independent of measures.

Triangle density varies with geometric configurations.

Abstract

We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $\mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz number (the conjectured crossing number of $K_{n,n}$). In this paper we consider the intersection graphs associated with such random drawings. We prove that for any probability measures, the resulting random intersection graphs form a convergent graph sequence in the sense of graph limits. The edge density of the limiting graphon turns out to be independent of the two measures as long as they are antipodally symmetric. However, it is shown that the triangle densities behave differently. We examine a specific random model, blow-ups of antipodal drawings $D$ of $K_{4,4}$, and show that the triangle density in the corresponding crossing graphon depends on the angles between the great circles containing the edges in $D$ and can attain any value in the interval $\bigl(\frac{83}{12288}, \frac{128}{12288}\bigr)$.