Matchings and paths in the cube (Q1363756)

The authors investigate lower bounds on the size of matchings from a subset of the veretx set of the \(n\)-cube to its complement, on the number of internally vertex-disjoint paths between two vertex-disjoint subsets of the \(n\)-cube, and on the number of edge-disjoint paths between two vertex-disjoint subsets of the \(n\)-cube. They present an intriguing conjecture on the number of arc-disjoint directed paths between a down-set and up-set which are vertex-disjoint, where all edges of the \(n\)-cube are oriented from a set towards supersets.