136 problems

For every number of points $n\ge 1$, integers $\ell,k\ge 1$ with $k>\ell$, and configurations $x_1,\ldots,x_n\in\mathbb{R}^{\ell}$ and $y_1,\ldots,y_n\in\mathbb{R}^{k}$, define … I…

For all positive integers $d$ and $t$, every $t d(d+1)$-connected graph contains $t$ pairwise edge-disjoint spanning subgraphs, each of which is $d$-rigid. More generally, for all…

For every pair of uniform matroids $U_{r,n}$ and $U_{s,m}$, there exists a freest matroid product of $U_{r,n}$ and $U_{s,m}$; equivalently, the relevant class of products has a max…

Dewar's conjecture. There exists a function $f \colon \mathbb{N}\rightarrow \mathbb{N}$ with the following property. For $d\in \mathbb{N}$, if $G$ is a vertex-transitive graph of d…

Let $d\geq 1$ and let $P_{n,d}$ be the generalized path graph, an $n$-vertex minimally $d$-rigid graph. For a graph $G$, let $a_d(G)$ denote its $d$-dimensional algebraic connectiv…

Let $d\geq 1$ and let $K_{2d+1}$ be the complete graph on $2d+1$ vertices. A realisation of $K_{2d+1}$ in $\ell_\infty^d$ assigns a point of $\mathbb{R}^d$ to each vertex, and it i…

Edge-bound conjecture. One has

Redundant symmetry-rigidity conjecture. For every such $\Gamma$ and $e$, $(G({\cal S})-e-e^*,*)$ is $\Gamma$-rigid in $\mathbb R^{k+1}$.

Minimal rigidity conjecture. If $(\Gamma,p)$ is minimally rigid, then $(\Gamma,p)$ is rigid.

Let $G$ be a graph with a triangulation $T$ of a surface $S$ as a spanning subgraph. A graph is globally rigid when its generic realization is uniquely determined, up to Euclidean…

Let $\langle H,m\rangle$ be a body-bar orbit framework on the fixed torus $\mathcal T_0^d$, and let $\mathcal G_{\mathcal C}(Y)$ denote the collection of connected components as…

Let $d\geq 1$, let $G$ be a graph, and let $F$ be a set of nonedges of $G$. For each $f\in F$, write $(G\cup(F\setminus\{f\},f))$ for the graph-nonedge pair obtained by adding the…

Let $P$ be a convex $n$-gon, and place $n$ distinct points on the moment curve in $\mathbb{R}^{4}$. The associated bar-and-joint rigidity matroid is represented by the rigidity mat…

Let $P$ be a convex $n$-gon, and let $k$-triangulations be its maximal sets of diagonals containing no $k+1$ mutually crossing diagonals. Place $n$ generic points on the moment cur…

Let $G=(V,E)$ be a rigid graph and let $u,v\in V$. A pair of vertices is globally linked if every generic realisation of $G$ determines the distance between its two vertices. Globa…

Graph characterization conjecture. Then $(G,p)$ is generically minimally rigid on ${\mathcal{M}}$ if and only if $G=K_{2}$, $K_3$, $K_4$, or $G$ is $(2,1)$-tight.

Let $G$ be the product of $n\geq 2$ copies of $\operatorname{SL}(2,\mathbb{R})$. Let $U_1$ and $U_2$ be the products of, respectively, upper- and lower-unipotent one-parameter subg…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group whose factors are all non-compact, let $\Gamma<G$ be an irreducible lattice, and let $\rho$ be an affine action of…

Let $G$ be a semi-simple Lie group with no compact factors and no simple factors isomorphic to $SO(1,n)$ or $SU(1,n)$, and let $\Gamma<G$ be a lattice. A generalized quasi-affine a…

Let $G$ be a plane graph. A plane graph is generically rigid if its generic realizations are infinitesimally rigid, and it can be straightened as a pseudo-triangulation if its embe…

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 $W$,

Let $S$ be a closed piecewise-linear surface, and let $(G_S^\star,\boldsymbol p)$ be its coned framework: the framework has the vertices and edges of $S$, together with a cone vert…

Let $S(t)$ be a differentiable one-parameter family of oriented piecewise-linear surfaces. For each facet $F$, let $n_F$ be its unit normal and $V_F$ its volume, and define the dih…

Let $d\geq 3$, and let $p:[n]\to\mathbb{R}^d$ satisfy … for all $i\in[n]$ and … Assume that the image of $p$ has size at least $3$. Lew–Noy–Plaza–Eberhardt's conjecture. The second…