16 problems

The partition characterisation conjecture. The graph $G$ is $(3,6)$-tight if and only if, for every edge $e\in E$, there exists a partition $(S_1,S_2,S_3)$ of $E$ such that

For a graph $G$, define … and let $g(n,d)$ be the smallest integer such that every $n$-vertex graph $G$ with $\eta(G)\geq g(n,d)$ is $d$-rigid. Degree-sum conjecture. If…

Let $f(n,d)$ be the smallest integer such that every $n$-vertex graph with minimum degree at least $f(n,d)$ is $d$-rigid. A graph is $d$-rigid when every generic framework in…

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

Stress-linked pair component conjecture. If $\{u,v\}$ is $d$-stress-linked in $G$, then there is some $\mathcal{R}_d$-component $G_0$ of $G$ such that $\{u,v\}$ is $d$-stress-linke…

Minimally $\mathcal{R}_d$-connected stress-independence conjecture. Every minimally $\mathcal{R}_d$-connected graph is $d$-stress-independent.

Stronger edge bound conjecture. The number of edges satisfies

Let $D$ be a double ${\mathcal R}_3$-circuit, and let $w$ be a technicolour vertex of degree $5$ in $D$. The principal partition of $D$ is the partition associated with its double-…

Let $L_1, \ldots, L_d \subset \mathbb{C}^m$ be linear spaces not contained in any coordinate hyperplane, and let $r_1, \ldots, r_d$ be the rank functions of the corresponding linea…

Jackson–Owen conjecture. Every minimally $2$-rigid graph $G$ with $n$ vertices satisfies

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…

Let $G$ be a planar graph, and let $e$ and $f$ be edges added to $G$. A graph is $(3,6)$-tight if it has $3|V|-6$ edges and every subgraph $H$ with at least three vertices satisfie…

Random-graph rigidity conjecture. For every $p=\omega(\log n/n)$, $G\sim G(n,p)$ is $\textbf{whp}$ $d$-rigid for

Let $G$ be a graph with at least one edge. An orientation of $G$ is acyclic if it has no directed cycles, and a cycle is stretched when it has the form…

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.