On Asymptotic Packing of Geometric Graphs

TL;DR

This paper investigates the asymptotic packing properties of geometric graphs, identifying which graphs can be packed into complete graph drawings and characterizing their packability in convex settings.

Contribution

It introduces the concept of geometric-packability and convex-packability, providing classifications for planar Hamiltonian graphs and explicit constructions for packings.

Findings

Triangles are geometric-packable due to Steiner Triple Systems.

4-cycle graphs are geometric-packable in plane drawings.

Most other planar Hamiltonian graphs are not geometric-packable.

Abstract

A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph $G$, we determine whether or not a plane $G$ is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of $K_n$.