25 problems

Combinatorial-zeolite conjecture. Every 6-connected 3-dimensional combinatorial zeolite is globally rigid in $\mathbb{R}^3$.

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…

Clique exclusion conjecture. The graph $G$ does not contain a copy of $K_{d+2}$.

Let $G=(V,E)$ be a graph. A framework is globally $\mathcal{R}_3$-rigid if every framework in $\mathbb{R}^3$ with the same edge lengths differs from it by a composition of isometri…

Full stress-rank conjecture. If $(G,p)$ is globally $(d,k)$-rigid and $G$ is not complete, then there exists $\sigma \in \ker DR_k(G,p)^T$ such that

Random 4-regular graph rigidity conjecture. A random $4$-regular graph with $n$ vertices is globally rigid in $\mathbb{R}$ with high probability.

Giro et al.'s reconstruction conjecture. For every $\varepsilon>0$ and every $p\geq(1+\varepsilon)/n$, with high probability there exists a subset $V'\subseteq[n]$ of size…

One-discus global-rigidity conjecture. The graph $G^\dagger$ is generically globally rigid in $\mathbb R^3$ if and only if it is $4$-connected and redundantly rigid in…

Let $G$ be a graph and let $u,v\in V(G)$ be a pair of vertices. Let $K(G,p)$ denote the shared stress kernel associated with a generic realization $(G,p)$. Suppose that $\{u,v\}$ i…

Let $G=G_1\cup G_2$, where $|V(G_1)\cap V(G_2)|\leq d+1$, and let $u,v\in V(G_1)\cap V(G_2)$. A pair is $\mathcal{R}_d$-linked when it is contained in an $\mathcal{R}_d$-circuit. J…

Let $p$ be an even positive integer with $p\neq 2$, let $n\geq 3$, and let $G$ be a connected graph with $n$ vertices. Write ${\cal CM}_n^p$ for the relevant $\ell_p$-configuration…

Gluing characterization conjecture. If $\{u,v\}$ is not linked in $G_2$ in $\mathbb{R}^d$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$ if and only if it is globally…

Gluing conjecture. If $\{u,v\}$ is linked in $G_i$ in $\mathbb{R}^d$ for $i\in\{1,2\}$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$.

Let $G$ be a graph, let $d$ be a positive integer, and let $uv\in E(G)$. Suppose that $G$ is $(d+1)$-connected, $|N_G(u)\cap N_G(v)|=d-1$, and $G/uv$ is globally rigid in…

Let $G$ be a globally rigid graph in $\mathbb{R}^d$, and let $e$ be an edge such that $G-e$ is not globally rigid. An edge is an $\mathcal{R}_{d+1}$-bridge when its deletion lowers…

Let $G=(V,E)$ be a graph. A pair of vertices is linked in $G$ in $\mathbb{R}^k$ when its distance is fixed across all generic equivalent realizations, and globally linked when that…

Let $d$ be a positive integer and let $G=(V,E)$ be a graph on at least $d+2$ vertices that is minimally globally rigid in $\mathbb{R}^d$. The extremal sparsity conjecture. … and th…

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 $S$ be a surface, let $(G,p)$ be a framework generic on $S$, and let $(\omega,\lambda)$ be an equilibrium stress for $(G,p)$. Write $\Omega_S(\omega,\lambda)$ for its stress ma…

Let $(G,p)$ be a generic framework on ${\mathcal{M}}$, where ${\mathcal{M}}\in\{{\mathcal{Y}},{\mathcal{C}}\}$. A framework is globally rigid on ${\mathcal{M}}$ when every framewor…

Let $G=(V,E)$ be a graph with $|V|\geq 5$, let ${\mathcal{M}}$ be a circular cylinder, and let $p$ be generic for ${\mathcal{M}}$. The global rigidity characterisation conjecture.…

Consider vertex splitting in a three-dimensional framework, with both the added vertex and the split vertex at least $4$-valent. Vertex-splitting circuit conjecture. The operation…

Consider a vertex splitting in a three-dimensional framework, with both the added vertex and the split vertex at least $4$-valent. Connelly–Whiteley vertex-splitting conjecture. Su…

Blow-up conjecture. For any connected graph $G$ with $v > 1$, there exists some $a_1, a_2, \ldots, a_v$ and some $d$ such that if we replace each $v_i$ with an independent set of s…

Connectivity conjecture. Any $(d+1)$-connected $k$-chain in $\mathbb{R}^d$ with more than $\binom{d+2}{2}$ vertices is generically globally rigid.