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

Abstract

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={s1,s2,…,sk} and T={t1,t2,…,tk} be two sets of k vertices in different partite sets of the n-dimensional hypercube Qn, and let e=|{i|siandtiare adjacent,1⩽i⩽k}|. If n>k+⌈(k-e)/2⌉, then there exist k vertex-disjoint paths P1,P2,…,Pk, where Pi connects si and ti, for i=1,2,…,k, such that these k paths contain all vertices of Qn.