ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS

Abstract

A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of unit balls in Euclidean d-space is called totally separable if any two unit balls can be separated by a hyperplane such that it is disjoint from the interior of each unit ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) proved upper bounds for the number of touching pairs (called contact number) in a totally separable packing of n unit balls in Euclidean dspace. In this paper we improve their upper bounds and extend them to the so-called locally separable packings. We give a sharp estimate in the plane. Here, 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.