Hamiltonicity of hypercubes with a constraint of required and faulty edges

Let R and F be two disjoint edge sets in an n-dimensional hypercube Q(n) . We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F. Besides, considering every vertex of Q(n) incident to at most n - 2 edges of F, we show that a Hamiltonian cycle exists if (A) vertical bar R vertical bar + 2 vertical bar F vertical bar <= 2n - 3 when vertical bar R vertical bar >= 2, or (B) vertical bar R vertical bar + 2 vertical bar F vertical bar <= 4n - 9 when vertical bar R vertical bar <= 1. Both bounds are tight. The analogous property for Hamiltonian paths is also given.