Identifying contact graphs of sphere packings with generic radii

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.

22 pages, 2 figures. Correction made to Lemma 3.1 (now Lemma 3.2) and Corollary 1.3 replaced by Conjecture 5.1

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