Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

Two Hamiltonian paths are said to be fully independent if the ith vertices of both paths are distinct for all i between 1 and n, where n is the number of vertices of the given graph. Hamiltonian paths in a set are said to be mutually fully independent if two arbitrary Hamiltonian paths in the set are fully independent. On the other hand, two Hamiltonian cycles are independent starting at v if both cycles start at a common vertex v and the ith vertices of both cycles are distinct for all i between 2 and n. Hamiltonian cycles in a set are said to be mutually independent starting at v if any two different cycles in the set are independent starting at v. The n-dimensional hypercube is widely used as the architecture for parallel machines. In this paper, we study its fault-tolerant property and show that an n-dimensional hypercube with at most n-2 faulty edges can embed a set of fault-free mutually fully independent Hamiltonian paths between two adjacent vertices, and can embed a set of fault-free mutually independent Hamiltonian cycles starting at a given vertex. The number of tolerable faulty edges is optimal with respect to a worst case.