# cover

## Strong matchings and covers ★★★

Author(s): Aharoni

Let be a hypergraph. A *strongly maximal* matching is a matching so that for every matching . A *strongly minimal* cover is a (vertex) cover so that for every cover .

**Conjecture**If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

## 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.

## Faithful cycle covers ★★★

Author(s): Seymour

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

## (m,n)-cycle covers ★★★

Author(s): Celmins; Preissmann

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

## Cycle double cover conjecture ★★★★

**Conjecture**For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.