Abstrakt
Two ordered Hamiltonian paths in the n-dimensional hypercube Q_n are said to be independent if ith vertices of the paths are distinct for every 1 <= i <= 2^n. Similarly, two s-starting Hamiltonian cycles are independent if the ith vertices of the cycle are distinct for every 2 <= i <= 2^n.
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 w_i and b_i, there are n mutually independent Hamiltonian paths with endvertices w_i, b_i in Q_n.
We also show that Q_n contains n - f fault-free mutually independent s-starting Hamiltonian cycles, for every set of f <= n - 2 faulty edges in Q_n 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.