Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

Abstract

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.

10 pages

References in corpus (1)

• On the separation profile of infinite graphs

Cited by in corpus (8)

• Parabolic theory of the discrete p-Laplace operator
• Apollonian Ball Packings and Stacked Polytopes
• On limits of Graphs Sphere Packed in Euclidean Space and Applications
• On the separation profile of infinite graphs
• Graph topologies induced by edge lengths
• Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube
• Distance Geometry for Kissing Spheres
• The Liouville and the intersection properties are equivalent for planar graphs