Spanning multi-paths in hypercubes

A set AA of vertices of a hypercube is called balanced if |{A∈A∣|A|≡0mod2}|=|{A∈A∣|A|≡1mod2}|. We prove that for every natural number n   there exists a natural number π1(n)π1(n) such that for every hypercube QQ with dim(Q)⩾π1(n)dim(Q)⩾π1(n) there exists a family {Pi}i=1n of pairwise vertex-disjoint paths PiPi between AiAi and BiBi for i=1,2,…,ni=1,2,…,n with V(Q)=⋃i=1nV(Pi) if and only if {Ai,Bi∣i=1,2,…,n}{Ai,Bi∣i=1,2,…,n} is a balanced set.