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

Abstract

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 fa parts per thousand currency signn-2 denotes the total number of permanent edge-faults in Q (n) for na parts per thousand yen4, 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.