Károly Bezdek: More on contact numbers of unit sphere 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. We give a brief survey on bounding the contact numbers of unit sphere packings in Euclidean d-space. 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.