133 problems

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…

Let $g\geq 2$, let $S_g$ be the closed orientable surface of genus $g$, and let a simple graph be called rigid in $S_g$ when it has a well-positioned realisation whose associated g…

Jackson–Tanigawa's rigidity conjecture. For $d\geq 3$ and $\mathcal{X}=\{K_{d+2},K_{d+2,d+2}\}$, $\mathcal{R}_n^d$ is the unique maximal $\mathcal{X}$-matroid, and…

Jackson–Tanigawa's hyperconnectivity conjecture. For $d\geq 1$ and $\mathcal{X}=\{K_{d+2},K_{d+1,d+1}\}$, $\mathcal{H}_n^d$ is the unique maximal $\mathcal{X}$-matroid, and…

Let $H$ be a multigraph, let $G_H$ be the body-hinge graph induced by $H$, and let $d\geq 3$. A $k$-superbrick is the graph-theoretic object defined in the source for the multigrap…

Let $\Gamma=(V,E)$ be a $(d,\ell)$-tight multigraph, with $\ell<\frac{4d+2}{3}$. Let $K_2^{2k-\ell}$ denote the graph with two vertices and $2k-\ell$ parallel edges, and let a $d$-…

Finiteness conjecture. There are only finitely many values of $n$ for which there exists a symmetric abstract $2$-rigidity matroid $\mathcal{M}_n\neq\mathcal{R}_n$ in which…

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 $d$ be a positive integer, and let $G$ be a graph on $n$ vertices. Write ${\mathcal{S}}_d(G)$ and ${\mathcal{H}}_d(G)$ for the completability and hyperconnectivity matroids in…

Let $d\geq 2$, let $n\geq 2d+2$, and let $K_n$ denote the complete graph on $n$ vertices. A graph is edge-redundantly rigid in a normed space if it has a framework that is infinite…

For each $3$-connected embedded graph $(G,\mathbf p)$ on the unit sphere with topological degree one and with every facial perimeter of length less than $2\pi$, define a partial or…

Let $G$ be a vertex $3$-connected planar graph embedded in the unit-radius sphere. Assume that each edge is a geodesic of length less than $pi$ and each face has total perimeter le…