Identifying contact graphs of sphere packings with generic radii

TL;DR

This paper proves Ozkan et al.'s conjecture for sphere packings with contact graphs of the form G ⊕ K2, showing such packings are stress-free with at most 3n-6 contacts, and characterizes when these graphs are contact graphs.

Contribution

It establishes the conjecture for a specific class of contact graphs and characterizes when these graphs correspond to generic radii sphere packings.

Findings

Sphere packings with contact graph G ⊕ K2 are stress-free.

Such packings have at most 3n-6 contacts.

G ⊕ K2 is a contact graph iff G is a penny graph with no cycles.

Abstract

Ozkan et al. conjectured that any packing of $n$ spheres with generic radii will be stress-free, and hence will have at most $3n-6$ contacts. In this paper we prove that this conjecture is true for any sphere packing with contact graph of the form $G \oplus K_2$, i.e., the graph formed by connecting every vertex in a graph $G$ to every vertex in the complete graph with two vertices. We also prove the converse of the conjecture holds in this special case: specifically, a graph $G \oplus K_2$ is the contact graph of a generic radii sphere packing if and only if $G$ is a penny graph with no cycles.