# nowhere-zero flow

## The intersection of two perfect matchings ★★

Author(s): Macajova; Skoviera

Conjecture   Every bridgeless cubic graph has two perfect matchings , so that does not contain an odd edge-cut.

Keywords: cubic; nowhere-zero flow; perfect matching

## Half-integral flow polynomial values ★★

Author(s): Mohar

Let be the flow polynomial of a graph . So for every positive integer , the value equals the number of nowhere-zero -flows in .

Conjecture   for every 2-edge-connected graph .

Keywords: nowhere-zero flow

## A nowhere-zero point in a linear mapping ★★★

Author(s): Jaeger

Conjecture   If is a finite field with at least 4 elements and is an invertible matrix with entries in , then there are column vectors which have no coordinates equal to zero such that .

Keywords: invertible; nowhere-zero flow

## Unit vector flows ★★

Author(s): Jain

Conjecture   For every graph without a bridge, there is a flow .

Conjecture   There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.

Keywords: nowhere-zero flow

## Real roots of the flow polynomial ★★

Author(s): Welsh

Conjecture   All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

## A homomorphism problem for flows ★★

Author(s): DeVos

Conjecture   Let be abelian groups and let and satisfy and . If there is a homomorphism from to , then every graph with a B-flow has a B'-flow.

Keywords: homomorphism; nowhere-zero flow; tension

## The three 4-flows conjecture ★★

Author(s): DeVos

Conjecture   For every graph with no bridge, there exist three disjoint sets with so that has a nowhere-zero 4-flow for .

Keywords: nowhere-zero flow

## Bouchet's 6-flow conjecture ★★★

Author(s): Bouchet

Conjecture   Every bidirected graph with a nowhere-zero -flow for some , has a nowhere-zero -flow.

Keywords: bidirected graph; nowhere-zero flow

## Jaeger's modular orientation conjecture ★★★

Author(s): Jaeger

Conjecture   Every -edge-connected graph can be oriented so that (mod ) for every vertex .

Keywords: nowhere-zero flow; orientation

## 3-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every 4-edge-connected graph has a nowhere-zero 3-flow.

Keywords: nowhere-zero flow