9 problems

A stable representation of a graph is an ${\mathcal{M}}$-representation that is a local minimum with respect to the ordering relation defining stability. Finite algorithm conjectur…

Aperiodic disk-packing conjecture. There exists a finite set of disk sizes that admits a triangulated packing other than the hexagonal compact packing, while no such packing is per…

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highes…

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the s…

Isostatic conjecture. The density $\rho$ of every such packing satisfies

For each integer $N\geq 2$, let $\epsilon(N)$ be a positive quantity with … as $N$ tends to infinity. Let $M$ be the Markov constant of a packing of $N$ equal disks in a square tor…

Let $N\geq 6$ and consider a rigid packing of $N$ equal disks in a square torus. Assume that its packing graph consists of two linear geodesics, and let $M$ be its Markov constant.…

Square-torus Markov-bound conjecture. Every such packing satisfies $M\leq 6.25$. Equivalently,

Let $N$ equal disks be packed in a square torus. A packing is said to correspond to a convergent of $1/\sqrt{3}$ or $2-\sqrt{3}$ when it is one of the Type I or Type II packings, r…