Uniform Geodesic Drawings of Graphs

TL;DR

This paper investigates crossing numbers in dense graph drawings with vertices uniformly distributed on spheres or convex domains, establishing inequalities and bounds that connect continuous and discrete models.

Contribution

It proves a sharp inequality for geodesic crossings on the sphere and a planar analogue, linking continuous results to finite graphs and confirming a conjectured crossing constant.

Findings

Minimizing crossings by connecting points within a fixed distance on the sphere

Established a planar analogue for straight-line drawings in convex domains

Confirmed the conjectured crossing constant lower bound in the small density limit

Abstract

We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a continuous setting: among all measurable edge arrangements of a fixed density, the amount of crossings is minimized by connecting pairs of points within a fixed distance threshold. We also prove a planar analogue for straight-line drawings in convex planar domains. We transfer these continuous results to finite graphs using a smoothing argument. In the small density limit, we recover the conjectured midrange crossing constant lower bound of $8/(9\pi^2)$ for this restricted model.