Paired many-to-many disjoint path covers of hypercubes with faulty edges

Let M"k={u"i,v"i}"i"="1^k be a set of k pairs of distinct vertices of the n-dimensional hypercube Q"n such that it contains k vertices from each class of bipartition of Q"n. Gregor and Dvorak proved that if n>2k, then there exist k vertex-disjoint paths P"1,P"2,...,P"k containing all vertices of Q"n, where two end-vertices of P"i are u"i and v"i for i=1,2,...,k. In this paper we show that the result still holds if removing n-2k-1 edges from Q"n. When k=2, we also show that the result still holds if removing 2n-7>=1 edges from Q"n such that every vertex is incident with at least three fault-free edges, and the number 2n-7 of faulty edges tolerated is sharp.