Identifying contact graphs of sphere packings with generic radii
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910285837631488 |
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| author | Dewar, Sean |
| author_facet | Dewar, Sean |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12588 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Identifying contact graphs of sphere packings with generic radii Dewar, Sean Combinatorics Metric Geometry 05B40 (Primary) 52C17, 52C25 (Secondary) 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. |
| title | Identifying contact graphs of sphere packings with generic radii |
| topic | Combinatorics Metric Geometry 05B40 (Primary) 52C17, 52C25 (Secondary) |
| url | https://arxiv.org/abs/2302.12588 |