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 S = {s(1), s(2),...,S-k} and T = {t(1), t(2),...,t(k)} be two sets of k vertices in different partite sets of the n-dimensional hypercube Q(n), and let e = vertical bar{i vertical bar s(i) and t(i) are adjacent,1 <= i <= k}vertical bar. If n>.k + left perpendicular(k - e)/2right perpendicular, then there exist k vertex-dis-joint paths P-1, P-2,...,P-k, where P-i connects s(i) and t(i), for i = 1,2,...,k, such that these k paths contain all vertices of Q(n). (C) 2013 Elsevier Inc. All rights reserved.