Contact graphs of ball packings

TL;DR

This paper establishes upper bounds on the average degree of contact graphs in ball packings across three to five dimensions, advancing understanding of geometric packing structures.

Contribution

It provides new upper bounds for the average degree of contact graphs in 3, 4, and 5-dimensional ball packings, including cases with varying radii.

Findings

Average degree in 3D packings ≤ 13.92

New bounds for 4D and 5D contact graphs

Extends geometric packing theory to higher dimensions

Abstract

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $\mathbb{R}^3$ is not greater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $\mathbb{R}^4$ and $\mathbb{R}^5$.