Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space
The contact graph of an arbitrary finite packing of 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 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. 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 sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n > 1 congruent balls in Euclidean 3-space, then for all with ka parts per thousand yen2.
contact number congruent sphere packing density (truncated) voronoi cell union of balls isoperimetric inequality spherical cap packing