Continuing the investigations of [1] and [2] we study the following two open problems. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on some classical problems on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then we wish to generate packings of n unit balls in such a special way that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding the largest possible contact number for the finite packings of n unit balls obtained in this way.

[1] K. Bezdek, On the maximum number of touching pairs in a finite packing of translates of a

convex body, J. Combin. Theory Ser. A 98/1 (2002), 192--200. [2] H. Harborth, Solution of Problem 664A, Elem. Math. 29 (1974), 14--15.