38 problems

Let $G$ be a bridgeless cubic graph. The $1$-dimensional flow number $\phi_1(G)$ is defined via nowhere-zero circular flows, equivalently $(r,1)$-NZFs. Tutte's 5-Flow Conjecture. E…

Let $G$ be a cubic graph admitting a nowhere-zero $4$-flow. An edge $e$ is $4$-removable if $G-e$ has a nowhere-zero $4$-flow. Hoffmann-Ostenhof's conjecture. Every cubic graph adm…

Let $d\ge 3$, and let $G$ be a bridgeless graph. A $(2,d)$-NZF is a $d$-dimensional nowhere-zero $2$-flow: an orientation of $G$ and a function from the edges to $\mathbb{R}^d$ who…

Let $M$ be a loopless oriented matroid with no $M(K_4)$ minor. A Hadwiger conjecture for oriented matroids. $M$ has a nowhere-zero $3$-coflow. This is the first non-trivial open ca…

Weak three-flow conjecture. There exists a fixed natural number $k$ such that every graph that is at least $k$-connected admits a nowhere-zero $3$-flow.

Esperet et al.'s flow reconfiguration conjecture. For each graph $G$ and each positive integer $k$,

Eulerian flow-connectedness conjecture. The following two claims hold:

Let $G$ be a graph, and let $\mathcal{F}(G,\mathbb{Z}_2^k)$ denote the reconfiguration graph of nowhere-zero $\mathbb{Z}_2^k$-flows. A nowhere-zero $\mathbb{Z}_2^k$-flow…

Let $G$ be a graph, and let $\mathcal{F}(G,A)$ denote its reconfiguration graph whose vertices are nowhere-zero $A$-flows, with adjacency given by changing flow values only on a cy…

Let $G$ be a graph, and let $\mathcal{F}(G,k)$ denote its reconfiguration graph whose vertices are nowhere-zero $k$-flows, with adjacency given by changing flow values only on a cy…

Unimodality conjecture for flow numbers. As a function of $p$, $\Phi_d^p(G)$ is unimodal: there exists $p^*\in[1,\infty]$ such that it is non-decreasing for $p\leq p^*$ and non-inc…

Half-flow-pair conjecture. Every bridgeless graph admits a $1/2$-flow-pair.

Two-dimensional Chebyshev flow-number conjecture. For every bridgeless graph $G$,

Xie and Zhang's conjecture. Every bridgeless graph admits a $(3,3)$-flow parity-pair-cover.

Archdeacon's conjecture. Every bridgeless graph has a $5$-OCDC.

Three-dimensional Manhattan flow conjecture. For every bridgeless graph $G$,

The 6-edge-connectivity conjecture. Every 6-edge-connected graph has a strongly connected modulo $3$-orientation.

Let $(\mathfrak{G},z)$ be a non-trivial flow-critical tame canvas, and let $C(\mathfrak{G},z)$ denote its associated degree sequence. If … and $\mathfrak{G}$ has a vertex of degree…

Let $G$ be a connected-flow-critical graph, meaning that $G$ does not admit a nowhere-zero $3$-flow and, for every non-trivial partition $\mathcal{P}$ of $V(G)$ whose parts each in…

Circular chromatic number conjecture. If $f(g) \geq 4g$, then the circular chromatic number of $G$ is at most $2+\frac{1}{g}$.

Group-flow conjecture. Every flow-admissible signed graph has a nowhere-zero $A$-flow for every abelian group $A$ with $|A|\geq 6$.

Let $G$ be a $3$-edge-connected Hamiltonian cubic graph, and let $fn(G)$ denote its Frank number. Hamiltonian cubic graph conjecture. Every $3$-edge-connected Hamiltonian cubic gra…

Let $G$ be a cyclically $4$-edge-connected graph, meaning that deleting fewer than $4$ edges cannot separate $G$ into two components both containing a cycle. Let $fn(G)$ denote its…

Let $G$ be a $3$-edge-connected graph. Its Frank number $fn(G)$ is the minimum number of orientations of $G$ such that every edge is deletable in at least one orientation. Hörsch–S…

Petersen graph flow-number conjecture.