Cycles passing through prescribed edges in a hypercube with some faulty edges

In this paper, we consider the problem of cycles passing through prescribed edges in an n-dimensional hypercube Qn with some faulty edges. We obtain the following result: Let n > h ≥ 2,F⊂E(Qn) with |F| < n-h, and E0⊂E(Qn\F with |E0|=h. If the subgraph induced by E0 consists of pairwise vertex-disjoint paths, then all edges of E0 lie on a cycle of every even length from 2h-1(n+1-h)+2(h-1) to 2n in the graph Qn-F. Moreover, if h=2, then the result is optimal in the sense that Qn contains (1) two edges such that any cycle in Qn passing through them is of length at least 2n, and (2) edge subsets E0 and F with |E0|=2,|F|=n-2 such that no Hamiltonian cycle passes through the two edges of E0 in Qn-F.