On contact numbers of locally separable unit sphere packings

TL;DR

This paper improves upper bounds on contact numbers for locally separable unit sphere packings in Euclidean space and characterizes maximum contact packings in the plane.

Contribution

It extends previous bounds from totally separable to locally separable packings and provides a crystallization characterization in two dimensions.

Findings

Improved upper bounds for contact numbers of locally separable packings.

Characterization of maximum contact number packings in the plane.

Extension of bounds from totally separable to locally separable packings.

Abstract

The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of balls in Euclidean $d$-space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) upper bounded the contact numbers of totally separable packings of $n$ unit balls in Euclidean $d$-space in terms of $n$ and $d$. In this paper we improve their upper bound and extend that new upper bound to the so-called locally separable packings of unit balls. We call a packing of unit balls a locally separable packing if each unit ball of the packing together with the unit balls that are tangent to it form a totally separable packing. In the plane, we prove a crystallization result by characterizing all locally separable packings of $n$ unit disks having maximum contact number.