On the mutually independent Hamiltonian cycles in faulty hypercubes

Two ordered Hamiltonian paths in the n-dimensional hypercube Qn are said to be independent if ith vertices of the paths are distinct for every 1 ⩽ i ⩽ 2n. Similarly, two s-starting Hamiltonian cycles are independent if the ith vertices of the cycle are distinct for every 2 ⩽ i ⩽ 2n. A set S of Hamiltonian paths (s-starting Hamiltonian cycles) are mutually independent if every two paths (cycles, respectively) from S are independent. We show that for n pairs of adjacent vertices wi and bi, there are n mutually independent Hamiltonian paths with endvertices wi, bi in Qn. We also show that Qn contains n − f fault-free mutually independent s-starting Hamiltonian cycles, for every set of f ⩽ n − 2 faulty edges in Qn and every vertex s. This improves previously known results on the numbers of mutually independent Hamiltonian paths and cycles in the hypercube with faulty edges.