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 Qn. 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 Qn incident to at most n − 2 edges of F , we show that a Hamiltonian cycle exists if (A) |R| + 2|F | ≤ 2n− 3 when |R| ≥ 2, or (B) |R| + 2|F | ≤ 4n− 9 when |R| ≤ 1. Both bounds are tight. The analogous property for Hamiltonian paths is also given.