# Matchings

## The Berge-Fulkerson conjecture ★★★★

**Conjecture**If is a bridgeless cubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .

Keywords: cubic; perfect matching

## The intersection of two perfect matchings ★★

**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

## Matchings extend to Hamiltonian cycles in hypercubes ★★

Keywords: Hamiltonian cycle; hypercube; matching

## Random stable roommates ★★

Author(s): Mertens

**Conjecture**The probability that a random instance of the stable roommates problem on people admits a solution is .

Keywords: stable marriage; stable roommates