Disjoint empty disks supported by a point set

Abstract

For a planar point-set P, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in P lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and Urabe recently conjectured that $${D(n) = \lceil n/2 \rceil}$$ . Here we show that $${D(n) \geq n/2 + n/236 - O(\sqrt{n})}$$ and thereby disprove this conjecture.