Die minimale 3-Nachbarnpackung kongruenter Kreise (Q791161)

Let P be a packing of congruent circles in the Euclidean plane. Two circles of the packing with a common boundary point are said to be neighbours. If each circle of P has exactly \(n(\geq 2)\) neighbours, then P is called an n-neighbour packing. It is easy to see that an n-neighbour packing does not exist for \(n>6\), and consists of infinitely many circles for \(n=4,5\) and 6. The main result of the present paper states that the minimal number of circles in a 3-neighbour packing is 16. The proof makes essential use of the properties of a graph associated with the packing. \{Reviewer's remark: L. Fejes Tóth showed that the density of any 5- neighbour packing of equal circles is at least \(\sqrt{3\pi}/7\) [Period. Math. Hungar. 4, 221-229 (1973; Zbl 0274.52015)], whereas 4-neighbour packings exist that have density 0.\}