44 problems

Let a nonsingular flow be a flow without stationary points on the 3-sphere $S^3$. Seifert conjecture. Every nonsingular flow on $S^3$ has a periodic orbit. The conjecture was dispr…

Jaeger–Linial–Payan–Tarsi's $\mathbb Z_3$-Group Connectivity Conjecture. Every $5$-edge-connected graph is $\mathbb Z_3$-connected.

Let $G$ be a graph, let $C$ be a circuit of $G$, and let $G/C$ denote the graph obtained by contracting $C$. A nowhere zero 4-flow on a graph is a 4-flow whose support contains eve…

Mixed-graph threshold orientation conjecture. There is a constant $k>2$ such that every mixed graph with minimum pseudo dicut size at least $k$ has an orientation $E^+$ satisfying

Let $M$ be a smooth connected compact manifold, and let a smooth flow be a smooth $\mathbb R$ action on $M$. The flow is globally hypoelliptic if its associated leafwise Laplacian…

A minimal flow on a compact space is a flow for which every orbit is dense. Gottschalk conjecture. The 3-sphere $S^3$ does not support a minimal flow. The source states this as a h…

Let $M$ be a connected $n$-manifold. A homeomorphism $f$ of $M$ is almost periodic when the subgroup of $\operatorname{Homeo}(M)$ that it generates has compact closure. Almost peri…

Equivalent flow formulation. Every bridgeless cubic graph has a nowhere-zero $\mathbb{Z}_2^3$-flow $f$ such that every component of $(V,E\setminus F)$ contains an even number of ve…

Let $\mathbb{S}^2$ be the unit $2$-sphere. Spherical coloring conjecture. There exists a map $q:\mathbb{S}^2 \rightarrow \{\pm 1, \pm 2, \pm 3, \pm 4\}$ such that antipodal points…

Let $G$ be a bridgeless cubic graph. An $\mathbb{S}^2$-flow is a unit vector flow with values on the unit sphere $\mathbb{S}^2$. Bobby's spherical flow conjecture. Every bridgeless…

Let $S$ be a surface. Robertson's surface conjecture. There is a constant $r$ such that every $2$-edge-connected graph embedded in $S$ with representativity at least $r$ has a nowh…

Let $G$ be a $2$-edge-connected graph embedded in the torus. Robertson's conjecture. If the representativity of the embedding is at least $3$, then $G$ has a nowhere-zero $4$-flow.…

Let $f^t$ be a $C^{\infty}$-flow on the torus without singularities. A quasiminimal is the topological closure of a recurrent trajectory; a closed trajectory or a fixed point is ca…

Fan's weighted refinement. If there is a $\mathbb{Z}_2\times\mathbb{Z}_2$-flow $f$ such that

For a positive integer $k$, let $c_3(k)$ denote the smallest constant such that every $c_3(k)$-edge-connected graph has a modulo-$k$-orientation, meaning an orientation in which th…

A $t$-$(v,k,\lambda)$ design $\mathcal{D}=(X,\mathcal{B})$ consists of a $v$-element point set $X$ and a collection of $k$-subsets of $X$, called blocks, such that every $t$-subset…

A Steiner triple system ${\rm STS}(v)$ is a $2$-$(v,3,1)$ design. A zero-sum $n$-flow of a design $\mathcal{D}=(X,\mathcal{B})$ is a map … such that the sum of the values over all…

Exponential flow-count conjecture. There exists a fixed constant $c>1$ such that, for every such $G$, $\text{?}$, and $f$, there are at least $c^n$ flows $\text{?}$ avoiding $f$.

Let $M$ be a compact metric space and let … be a continuous flow. A flow $(\phi_t)_{t\in\mathbb R}$ is weakly expansive if it satisfies the weak expansiveness condition, and it is…

A flow on a connected space $X$ with countably many chain-components, all of which are topologically transitive, is said to have no constant of motion if every continuous function…

Neumann-Lara's conjecture. Every loopless planar digraph admits an NL-$2$-coflow.

Let a flow, equivalently a vector field, be given on a manifold. A singular hyperbolic flow is one whose invariant sets admit the corresponding singular-hyperbolic structure; a hom…

Open-and-dense sink/source conjecture. There exists an open and dense family of continuous flows or smooth semiflows on manifolds for which sectional asymptotic expansion or contra…

Let $G$ be a graph, and let $E(G)$ denote its edge set. A 3-flow on $G$ is a flow whose values lie in a group of order three, and its support is the set of edges on which the flow…

Let $G$ be a graph and let $z_0\in V(G)$. A graph is $\langle\mathbb{Z}_3\rangle$-extendable at $z_0$ if every pre-orientation of the edges incident with $z_0$ whose out-degree min…