On the Existence of Hamiltonian Circuits in Faulty Hypercubes

The problem of finding Hamiltonian circuits in faulty hypercubes is explored. There are many different Hamiltonian circuits in a nonfaulty hypercube. The question of interest here is the following: if a certain number of links are removed from the hypercube, will a Hamiltonian circuit still exist? In partial answer to this question are the following results. First, it is shown that for any n-cube $( n\geqq 3 )$ with $\leqq 2n - 5$ link faults in which each node is incident to at least two nonfaulty links, there exists a Hamiltonian circuit consisting of only nonfaulty links. Since as will be shown, there exists an n-cube with $2n - 4$ faulty links, in which each node is incident to at least two nonfaulty links, for which there is no Hamiltonian circuit, this result is optimal. Second, it is shown that the problem of determining whether an n-cube with an arbitrary number of link faults has a Hamiltonian circuit is NP-complete.