4 problems

Let $n\ge 2$ and consider a packing of $n$ circular disks whose radii all lie in an interval $[1-\epsilon,1]$. Stability conjecture. There exists an $\epsilon>0$ such that, for eve…

A graph is a generic-radii sphere-packing contact graph if it is the contact graph of a sphere packing whose radii are generic. NP-hardness conjecture. Determining whether a graph…

Let $G=(V,E)$ be the contact graph of a sphere packing $P$ with generic radii, where generic means that the radii form an algebraically independent set. Regard $P$ as a bar-and-joi…

Let ${\mathcal P}_r$ be a packing of disks of radius $r>0$ in the hyperbolic plane $\mathbb{H}^{2}$, and let ${\rm \deg}_{\rm avr}({\mathcal P}_r)$ denote the average degree of its…