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Mathematics > Combinatorics
Title: Identifying contact graphs of sphere packings with generic radii
(Submitted on 24 Feb 2023 (v1), last revised 3 Jan 2024 (this version, v2))
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.
Submission history
From: Sean Dewar PhD [view email][v1] Fri, 24 Feb 2023 11:54:00 GMT (18kb)
[v2] Wed, 3 Jan 2024 12:04:36 GMT (86kb,D)
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