Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if \{A is an element of A\\A\ = 0 mod 2}\ = \{A is an element of A \\A\ = 1 mod 2}\. We prove that for every natural number n there exists a natural number pi(1)(n) such that for every hypercube Q with dim(Q) >= pi(1)(n) there exists a family {P-i}(i=1)(n) of pairwise vertex-disjoint paths P-i between A(i) and B-i for i = 1, 2, n with V(Q) = boolean OR(n)(i=1) V(P-i) if and only if {A(i), B-i\i = 1, 2,..., n} is a balanced set. (c) 2006 Elsevier B.V. All rights reserved.