Path partitions of hypercubes

A path partition of a graph G is a set of vertex-disjoint paths that cover all vertices of G. Given a set P={{a"i,b"i}}"i"="1^m of pairs of distinct vertices of the n-dimensional hypercube Q"n, is there a path partition {P"i}"i"="1^m of Q"n such that a"i and b"i are endvertices of P"i? Caha and Koubek showed that for 6m==3, there is a balanced set P in Q"n such that 2m-e=n, but no path partition with endvertices prescribed by P exists.