Paired many-to-many disjoint path covers of the hypercubes

In this paper we consider the problem of paired many-to-many disjoint path covers of the hypercubes and obtain the following result. Let image and image be two sets of k vertices in different partite sets of the n-dimensional hypercube image, and let image. If image, then there exist k vertex-disjoint paths image, where image connects image and image, for image, such that these k paths contain all vertices of image.