ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS

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 et al . ( Discrete Math . 339(2) (2016), 668–676) 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.