# Recent Activity

## Jorgensen's Conjecture ★★★

Author(s): Jorgensen

Conjecture   Every 6-connected graph without a minor is apex (planar plus one vertex).

Keywords: connectivity; minor

## Highly connected graphs with no K_n minor ★★★

Author(s): Thomas

Problem   Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?

Keywords: connectivity; minor

## The Alon-Tarsi basis conjecture ★★

Author(s): Alon; Linial; Meshulam

Conjecture   If are invertible matrices with entries in for a prime , then there is a submatrix of so that is an AT-base.

## The permanent conjecture ★★

Author(s): Kahn

Conjecture   If is an invertible matrix, then there is an submatrix of so that is nonzero.

Keywords: invertible; matrix; permanent

## The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

Conjecture   For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.

## 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

## Partitioning edge-connectivity ★★

Author(s): DeVos

Question   Let be an -edge-connected graph. Does there exist a partition of so that is -edge-connected and is -edge-connected?

Keywords: edge-coloring; edge-connectivity

## Acyclic edge-colouring ★★

Author(s): Fiamcik

Conjecture   Every simple graph with maximum degree has a proper -edge-colouring so that every cycle contains edges of at least three distinct colours.

Keywords: edge-coloring

## Packing T-joins ★★

Author(s): DeVos

Conjecture   There exists a fixed constant (probably suffices) so that every graft with minimum -cut size at least contains a -join packing of size at least .

Keywords: packing; T-join

## Decomposing eulerian graphs ★★★

Author(s):

Conjecture   If is a 6-edge-connected Eulerian graph and is a 2-transition system for , then has a compaible decomposition.

Keywords: cover; cycle; Eulerian

## Faithful cycle covers ★★★

Author(s): Seymour

Conjecture   If is a graph, is admissable, and is even for every , then has a faithful cover.

Keywords: cover; cycle

## (m,n)-cycle covers ★★★

Author(s): Celmins; Preissmann

Conjecture   Every bridgeless graph has a (5,2)-cycle-cover.

Keywords: cover; cycle

## The circular embedding conjecture ★★★

Author(s): Haggard

Conjecture   Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.

Keywords: cover; cycle

## 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

## 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

## 5-flow conjecture ★★★★

Author(s): Tutte

Conjecture   Every bridgeless graph has a nowhere-zero 5-flow.

Keywords: cubic; nowhere-zero flow