arXiv2023
Let $\mathcal{P}$ be a packing of circular disks of radius $ρ>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leqλ\leqρ$. We say that $\mathcal{P}$ is a $λ$-separable packing of circular disks of radius $ρ$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $λ$ form a totally separable packing, i.e., any two disks of $\mathcal{P'}$ can be separated by a line which is disjoint from the interior of every disk of $\mathcal{F'}$. This notion bridges packings of circular disks of radius $ρ$ (with $λ=0$) and totally separable packings of circular disks of radius $ρ$ (with $λ=ρ$). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to $λ$-separable packings of circular disks of radius $ρ$ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of $λ$-separable packings of unit disks in the Euclidean plane are sharp for all $0\leqλ\leq 1$ with the extremal values achieved by $λ$-separable lattice packings of unit disks. On the other hand, the bounds of similar results in the spherical and hyperbolic planes are not sharp for all $0\leqλ\leqρ$ although they do not seem to be far from the relevant optimal bounds either. The proofs use local analytic and elementary geometry and are based on the so-called refined Molnár decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest.