K5 and K3,3 are Toroidal Penny Graphs
C\'edric Lorand
TL;DR
This paper explores the relationship between Penny Graphs and sphere packing on the flat torus, proving that certain complete graphs are representable as penny graphs on a torus.
Contribution
It demonstrates that the complete graphs K5 and K3,3 can be realized as penny graphs on the flat square torus, connecting graph theory with geometric packing problems.
Findings
K5 is a toroidal penny graph.
K3,3 is a toroidal penny graph.
Links between Penny Graphs and sphere packings are established.
Abstract
In this article we emphasize on the connection between two fields of study that are Penny Graphs, and the Optimal Packing of Spheres on the Flat Torus. We give a brief litterature overview on related results in the fields of planar graphs, penny graphs, toroidal penny graphs and spherical codes.We also show that and are penny graphs on the flat square torus.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph theory and applications · Limits and Structures in Graph Theory