Identifying contact graphs of sphere packings with generic radii

TL;DR

This work proves a sharp characterization for sphere packings with generic radii whose contact graphs have the form $G\oplus K_2$. The authors develop a three‑step strategy: (i) reduce to a 2D setting via Möbius transforms while preserving generic radii, (ii) show that a generic radii packing with $G\oplus K_2$ forces $G$ to be a forest, and (iii) establish that when $G$ is a forest and a penny graph, there exists an open set of packings realizing $G\oplus K_2$ with generic radii, yielding stress‑freeness. Consequently, a sphere packing with generic radii and contact graph $G\oplus K_2$ is stress‑free, and such a graph occurs exactly when $G$ is a forest and penny graph; the paper also derives corollaries on clique bounds and chordal cases, and discusses the problem’s computational complexity, conjecturing NP‑hardness of the general decision problem. Overall, the approach combines Möbius transformations, algebraic function arguments, and rigidity theory to connect 3D sphere packings with 2D penny graph realizations for this special graph family.

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.

Figures (2)

• Figure 1: The standard form sphere packing, with two spheres of identical size (the top and bottom spheres) in contact with all other spheres.
• Figure 2: One example of a framework described in \ref{['l:specialframework']} for $k=4$. Each vertex $v_i$ is positioned at the point $p_{v_i} = (x_i,y_i)$ in $\mathbb{R}^2$. Every edge $v_i v_{i+1}$ with $i \in \{1,2,3\}$ has length $r_i + r_j$, and every edge $v_0 v_i$ with $i \in \{1,2,3,4\}$ has length $\sqrt{r_i^2 + 2r_i}$. By \ref{['l:specialframework']}, the set $\{r_1,r_2,r_3,r_4\}$ is algebraically dependent if the edge $v_1v_4$ has length $r_1+r_4$.

Theorems & Definitions (39)

• Conjecture 1.1: Genericity Conjecture for Sphere Packings
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• Theorem 2.3: congortheran2019
• Theorem 2.4
• Corollary 2.5
• Lemma 2.6
• proof
• Lemma 3.1