25 problems

Let $(\Gamma,g,G)$ be an abelian gain graph, where $G$ is a torsion-free abelian group, possibly $2$-divisible. Let $Z_1(\Gamma)$ be the integral cycle group, and suppose it has a…

Let $\Gamma$ be a finite group and let $\mathcal{G}$ be a minor-closed class of $\Gamma$-gainable biased graphs. An excluded minor for $\mathcal{G}$ is a biased graph outside…

Let $\Gamma$ be a finite group. A $\Gamma$-gainable biased graph is a biased graph admitting a gain assignment over $\Gamma$, and an excluded minor for a class is a graph outside t…

Let $r$ and $s$ be odd, and let $\ell$ be odd. For a complex unit gain dumbbell graph with gain parameters $(\alpha,\beta)$ and energy $E(\Phi)$, the conclusion that the minimum of…

Let $\Gamma$ be a Frobenius group with Frobenius kernel $\Gamma_1$, and let $\mathcal{A}$ be a set of Frobenius complements. A sequence of universal models for a class…

Let $(G,\mathcal{B})$ be a biased graph, let $M$ be an elementary lift of the frame matroid $F(G,\mathcal{B})$, and suppose that $M$ is $3$-connected. Let $\mathbb{F}$ be a field a…

Let $\mathbb{F}$ be a field, let $G$ be a loopless graph with orientation $D$, and let $\psi$ and $\psi'$ be $(\mathbb{F}^+\rtimes \mathbb{F}^{\times})$-gain functions for $G$. Sup…

Let $G$ be a graph, let $\mathcal{B}$ be a bias set, and let $F(G,\mathcal{B})$ and $L(G,\mathcal{B})$ denote the associated frame and lifted-graphic matroids. Representations of t…

Let $\Gamma$ be a finite group, let $\Gamma_1$ be a subgroup, and let $\mathcal{A}$ be a set of groups. Finite excluded-minor conjecture. The class…

Let $\Gamma$ be a group, let $\Gamma_1$ be a subgroup, and let $\mathcal{A}$ be a set of groups such that $\Gamma/\Gamma_1$ is isomorphic to a group in $\mathcal{A}$. Infinite excl…

Extension conjecture. The proofs of this paper should extend to all odd-order cyclic groups.

Let the arrangements be the arrangements between the extended Catalan arrangements $\operatorname{Cat}(\ell,m)$ and $\operatorname{Cat}(\ell,m-1)$ described by the gain-$m$ edges,…

Let $G$ be a simple graph, and let $\mathcal{T}_G$ denote the collection of all $\mathbb{T}$-gain graphs on $G$. Two gain graphs are equienergetic if they have the same energy. The…

Walsh's finite-group gain-graph conjecture. The same conclusion holds for every nontrivial finite group $Gamma$.

Let $\Gamma$ be a finite abelian group of order at least $2$, and let $\mathbb{F}$ be a finite field. A $\Gamma$-graphic delta-matroid is a delta-matroid associated with a $\Gamma$…

Weak flexible 2-periodic placement characterization. There exists a full placement-lattice $(p,L)$ of $G$ in $\mathbb{R}^2$ such that $(G,p,L)$ is a flexible full $2$-periodic fram…

Type 3 exclusion conjecture. Then $G$ has either a type 1 or type 2 flexible 2-lattice NBAC-colouring, a fixed lattice NBAC-colouring, or

Type 3 realization conjecture. If $G$ has a type 3 flexible 2-lattice NBAC-colouring, then there exists a full placement-lattice $(p,L)$ of $G$ in $\mathbb{R}^2$ such that…

Flexible 2-periodic placement characterization. There exists a full placement-lattice $(p,L)$ of $G$ in $\mathbb{R}^2$ such that $(G,p,L)$ is a flexible full $2$-periodic framework…

Let $\mathbb F$ be a field, let $M$ be a frame matroid or lifted graphic matroid, and let $A$ be an $\mathbb F$-representation of $M$. A canonical representation is a frame matrix…

Let $\mathcal{C}$ denote the cone. Let $S$ be either the group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis, or the group…

Let $\mathcal{Y}$ denote the cylinder. Let $S$ be one of the groups $\mathcal{C}_{mh}$, $\mathcal{C}_{mv}$ or $\mathcal{S}_{2m}$, with the rotational axis equal to the cylinder axi…

Let $\mathcal{Y}$ denote the cylinder, and let $S$ be the cyclic group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis. Let $(G,p)$ be…

Let $\mathcal{S}$ denote the sphere. For a $S$-symmetric framework, let $S$ be either the group $\mathcal{C}_{mh}$ with $m$ even or the group $\mathcal{S}_{2m}$ with $m$ odd. Let…

Odd-rotation conjecture. The framework $(G,\mathbf{p})$ is infinitesimally rigid if and only if $G$ is 2-rigid.