Hamiltonian cycles in hypercubes with faulty edges

Let F   be a set of faulty edges in hypercube QnQn with |F|⩽3n-8|F|⩽3n-8 for n⩾5n⩾5. We prove that there still exists a fault-free Hamiltonian cycle in QnQn if the following two conditions are satisfied: (1) the degree of every vertex is at least two, and (2) there do not exist a pair of nonadjacent vertices in a 4-cycle whose degrees are both two after faulty edges are removed.