Identifying contact graphs of sphere packings with generic radii

2023/02/24 by Dewar, Sean
#05B40 (Primary) 52C17 #52C25 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2302.12588

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 ⊕ K₂, 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 ⊕ K₂ is the contact graph of a generic radii sphere packing if and only if G is a penny graph with no cycles.

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