21 problems

Orientable 5-cycle double cover conjecture. Every bridgeless graph admits an orientable $5$-cycle double cover.

Confluent-edge-free flow conjecture. Every rich flow admissible graph with maximum degree $\Delta$ has a nowhere-zero $O(\ln(\Delta))$-flow containing no pair of confluent edges.

Rich-flow conjecture for 3-edge-connected graphs. Every $3$-edge-connected graph with maximum degree $\Delta$ admits a rich $(\Delta+3)$-flow.

Rich flow number conjecture. If $\Delta\geq 5$, then $G$ admits a rich $\lfloor 1.5\Delta+1\rfloor$-flow.

A signed graph is flow-admissible if it admits a nowhere-zero flow. Bouchet's 6-flow conjecture. Every flow-admissible signed graph has a nowhere-zero 6-flow. This conjecture exten…

Let $G$ be a bridgeless cubic graph containing a Hamiltonian path, and let $\overline{F}$ denote a $2$-factor of $G$. A non-conflicting nowhere-zero $Z_2\times Z_2$-flow with respe…

Let $G$ be a cubic graph and let $F$ be a perfect matching; write $\overline{F}$ for the complementary $2$-factor. A non-conflicting nowhere-zero $Z_2\times Z_2$-flow with respect…

Let $G$ be a 3-edge-connected cubic graph different from the Petersen graph. A nowhere-zero $Z_2\times Z_2\times Z_2$-flow is a flow $f$ whose edge values are nonzero elements of…

Conjecture on valid orientations for two specified faces. Then $G$ has a valid orientation.

Conjecture on valid orientations with four exceptional boundary vertices. Then $G$ has a valid orientation.

The subcontraction conjecture. If $G$ is a facially $3$-colorable graph which does not have a subcontraction isomorphic to $K_{3,n}^+$ for some $n\geq 4$, then $G$ is $3$-flowable.

Stefán's class 1 circular-flow bound conjecture.

Stefán's circular-flow infimum conjecture.

For every positive integer $k$, let $G_{2k+1}$ be the Goldberg snark on $8(2k+1)$ vertices, and let $\Phi_c(G_{2k+1})$ denote its circular flow number. The Goldberg snark circular-…

Let $G$ be a 3-edge-connected cubic graph different from the Petersen graph $P_{10}$. A nowhere-zero $\mathbb{Z}_2^3$-flow is a flow $f:E(G)\to\mathbb{Z}_2^3\setminus\{0\}$ satisfy…

Let $G$ be a simple cubic graph, and let $bed(G)$ denote the set of flow parameters admitted by $G$ in the source. $(4\frac14,\frac14)$-flow conjecture. For every simple cubic grap…

Let $G$ be a cubic graph that admits a perfect matching. An orientable $4$-weak bisection is a $4$-weak bisection with the orientability property used in the source; a…

Let $G$ be a graph. A zero-sum $k$-flow of $G$ is an edge labeling with labels in $\{\pm1,\ldots,\pm(k-1)\}$ such that the sum of the labels on all edges incident with every vertex…

Let $\Gamma$ be a $3$-edge-colorable cubic graph. An embedding of $\Gamma$ in a surface is an embedding whose genus is the genus of $\Gamma$, and a nowhere-identity dihedral $2$-fl…

Let $\vec{G}$ be a directed graph. An antisymmetric $B$-flow is a $B$-flow in which no two arcs receive inverse elements of $B$; in particular, it is nowhere-zero when $0=-0$. Thom…

Let $G$ be a finite undirected graph. A zero-sum flow assigns a nonzero integer to each edge so that the sum of the assignments on all edges incident with every vertex is zero. A z…