Generation of Optimal Packings from Optimal Packings

We define two notions of generation between the various optimal packings Qm of m congruent disks in a subset K of R2. The first one that we call weak generation consists in getting Qn by removing m − n disks from Qm and by displacing the n remaining congruent disks which grow continuously and do not overlap. During a weak generation of Qn from Qm, we consider the contact graphs G(t) of the intermediate packings, they represent the contacts disk-disk and disk-boundary. If for each t, the contact graph G(t) is isomorphic to the largest common subgraph of the two contact graphs of Qn and Qm, we say that the generation is strong. We call strong generator in K, an optimal packing Qm which generates strongly all the optimal Qk with k < m. We conjecture that if K is compact and convex, there exists an infinite sequence of strong generators in K. When K is an equilateral triangle, this conjecture seems to be verified by the sequence of hexagonal packings Q∆(k) of ∆(k) = k(k + 1)/2 disks. In this domain, we also report that up to n = 34, the Danzer graph of Qn is embedded in the Danzer graph of Q∆(k) with ∆(k − 1) ≤ n < ∆(k). When K is a circle, the first five strong generators appears to be the hexagonal packings defined by Graham and Lubachevsky. When K is a square, we think that our conjecture is verified by a series of packings proposed by Nurmela and al. In the same domain, we give an alternative conjecture by considering another packing pattern.