Path bundles on n-cubes

A path bundle is a set of 2a2a paths in an n  -cube, denoted QnQn, such that every path has the same length, the paths partition the vertices of QnQn, the endpoints of the paths induce two subcubes of QnQn, and the endpoints of each path are complements. This paper shows that a path bundle exists if and only if n>0n>0 is odd and 0⩽a⩽n-⌈log2(n+1)⌉0⩽a⩽n-⌈log2(n+1)⌉.