6 problems

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 $P$ be a packing of $n$ spheres whose radii are generic, meaning algebraically independent. A bar-and-joint framework is stress-free when it admits no nonzero self-stress. Ozka…

Disc-packing conjecture. Any $(2,3)$-sparse graph is the contact graph of a disc packing with algebraically independent radii; equivalently, any $(2,3)$-sparse graph is the contact…

For a fixed integer $d\geq 2$, consider the problem of deciding whether a given graph is the contact graph of a packing of unit spheres in $\mathbb{E}^{d}$. Hliněný's conjecture. T…

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…