24 problems

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

Tutte's three flow conjectures. The following three statements all hold:

Let $G$ be a bridgeless graph, and let $\phi_{3,2}(G)$ be its three-dimensional Euclidean flow index. Jain's $S^2$-flow conjecture. … Equivalently, every bridgeless graph admits a…

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 $G$,

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 $2+\frac{1}{g}$.

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…

Petersen graph flow-number conjecture.

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…

For an abelian group $\Gamma$, a graph $G$ is $\Gamma$-connected if $(G,\beta)$ has a nowhere-zero flow for every $\Gamma$-boundary $\beta$. Jaeger et al.'s group-connectivity conj…

A graph $G$ is $\mathbb{Z}_3$-flow-critical if it is flow-critical with respect to nowhere-zero $\mathbb{Z}_3$-flows, as defined in the source. Density conjecture for…

Degree-sensitive density conjecture. For every $3$-flow-critical graph $G$ on $n$ vertices,

Let $G$ be a cyclically $4$-edge connected cubic graph. For a cycle $C$ of $G$, let $G/E(C)$ denote the graph obtained by contracting the edges of $C$, and let a nowhere-zero $4$-f…

Jaeger–Linial–Payan–Tarzi conjecture. For every prime $p$, there exists a constant $k_p$ such that the union, with repetitions, of any $k_p$ bases for $\mathbb{Z}_p^n$ forms an add…

Let $t>1$ be an integer and let $G$ be a $(2t+1)$-graph, meaning a graph in which every vertex has degree $2t+1$. The odd-regular flow conjecture. Every such graph satisfies … The…