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…

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 $P$) be the Petersen graph, and let $\phi_2(P)$ denote its $2$-dimensional flow number, the infimum of the real numbers $r$ for which $P$ has a $(r,2)$-dimensional nowhere-zero…

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 $G$ be a graph. A nowhere-zero $\mathbb{Z}_3$-flow on $G$ is a flow with values in $\mathbb{Z}_3$ that is nonzero on every edge; $G$ is 4-edge-connected if deleting fewer than…

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 $p$ be a prime number, let $n$ be a positive integer, and let $B_1,\ldots,B_t$ be linear bases of the vector space $\mathbb{Z}_p^n$. An additive basis is a multiset of vectors…

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.

Let $G$ be a bridgeless cubic graph, and let a $(3,3,3)$-flows double cover mean the flow-based double-cover structure described in the source. (3,3,3)-flows conjecture. Every brid…

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…

Let $n\beq 7$, and let a $\mathbb{Z}_3$-flow-critical graph be a connected graph with no nowhere-zero $\mathbb{Z}_3$-flow such that contracting any edge produces a graph with a now…

Let $G$ be a connected graph, let $\Gamma$ be a ribbon graph associated with $G$, and let $n$ be a power of $2$. Write $T(\Gamma,n,t)$ for the total face color polynomial and let…

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.

Conjecture on valid orientations with four degree-3 vertices. Such graphs have a valid orientation meeting every given valid prescription function.

Strong 3-flow conjecture. Every $5$-edge-connected graph is $\mathbb{Z}_3$-connected.

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-…