Disjoint empty disks supported by a point set

TL;DR

This paper disproves a conjecture about the maximum number of pairwise-disjoint empty disks supported by any n-point set in the plane, providing a new lower bound that exceeds the previously conjectured value.

Contribution

The authors establish a new lower bound for D(n), disproving the conjecture that D(n) equals ceiling of n/2, thus advancing understanding of geometric disk packings.

Findings

D(n) is at least n/2 + n/236 - O(√n)

The conjecture D(n)=⌈n/2⌉ is false

New bounds improve understanding of empty disk packings

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.