28 problems

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.…

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

Mixed-graph orientation conjecture. There is a constant $k>2$ such that for every mixed graph $M=(V,E\cup A)$, there is an orientation $E^+$ such that

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

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…

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…

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 $f$.

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…

Let a flow, equivalently a vector field, be given on a manifold. A flow is hyperbolic if its chain recurrent dynamics are hyperbolic; a singular cycle is a cycle involving singular…

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…

Mean-dimension embedding conjecture. If

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…

Let $G$ be a finite graph without loops, possibly with multiple edges. A graph is $\mathbb{Z}_3$-connected if, for every zero-sum function $\beta:V(G)\to\mathbb{Z}_3$, there is an…

Let $G$ be a bridgeless cubic graph. A 5-line Fano-flow is a Fano-flow in which at most five lines of the Fano plane occur as flow values at vertices. Five-line Fano-flow conjectur…

0-sum 3-flow conjecture. Every $2$-edge connected $5$-regular graph admits a $0$-sum $3$-flow.

Facet-connectivity reduction conjecture. If every $k$-facet-connected complex $\Delta$ has a $q$-flow for some $q<\kappa$, then every $(k-1)$-facet-connected complex $\Delta$ also…

Linear simplicial Tutte flow conjecture.

Simplicial Tutte flow conjecture. For every dimension $d$ there exists a number $\kappa(d)<\infty$ such that every bridgeless complex of dimension $d$ has a nowhere-zero $q$-flow f…

Rohlin-property characterization conjecture. The following three conditions are equivalent: