On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube

Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q _n contains (n?1?f)-mutually independent fault-free Hamiltonian cycles, where f≤n?2 denotes the total number of permanent edge-faults in Q _n for n≥4, and edge-faults can occur everywhere at random. Later, Kueng et al. (2009a) presented a formal proof to validate Hsieh and Yu’s argument. This paper aims to improve this mentioned result by showing that up to (n?f)-mutually independent fault-free Hamiltonian cycles can be embedded under the same condition. Let F denote the set of f faulty edges. If all faulty edges happen to be incident with an identical vertex s, i.e., the minimum degree of the survival graph Q _n ?F is equal to n?f, then Q _n ?F contains at most (n?f)-mutually independent Hamiltonian cycles starting from s. From such a point of view, the presented result is optimal. Thus, not only does our improvement increase the number of mutually independent fault-free Hamiltonian cycles by one, but also the optimality can be achieved.