Fault-Tolerant Embeddings of Hamiltonian Circuits in k-ary n-Cubes

We consider the fault-tolerant capabilities of networks of processors whose underlying topology is that of the k-ary n-cube $Q_n^k$, where $k\geq 3$ and $n\geq 2$. In particular, given a copy of $Q_n^k$ where some of the interprocessor links may be faulty but where every processor is incident with at least two healthy links, we show that if the number of faults is at most 4n-5, then $Q_n^k$ still contains a Hamiltonian circuit, but that there are situations where the number of faults is 4n-4 (and every processor is incident with at least two healthy links) and no Hamiltonian circuit exists. We also remark that given a faulty $Q_n^k$, the problem of deciding whether there exists a Hamiltonian circuit is NP-complete.