5 problems

Six-connectivity conjecture. Every 6-connected $K_4$-covered graph is rigid in $\mathbb{R}^3$.

Kiraly–Tanigawa's body-pin conjecture. The graph $G_H$ is rigid in $\mathbb{R}^3$ if and only if, for every partition $\mathcal{P}=\{P_1,P_2,\dots,P_t\}$ of $W$,

Whiteley's conjecture. These matroids are equal for every $n\geq 2$:

The Dress Conjecture. The rank of $G$ satisfies

A $k$-triangulation is a maximal set of pairwise noncrossing diagonals of a convex polygon in which no $k+1$ diagonals mutually cross. A graph is generically minimally rigid in dim…