How robust is the n-cube?

The n-cube network is called faulty if it contains any faulty processor or any faulty link. For any number k we are interested in the minimum number f(n, k) of faults, necessary for an adversary to make any (n-k)-dimensional subcube faulty. Reversely formulated: The existence of a (n-k)- dimensional nonfaulty subcube can be guaranteed, unless there are at least f(n,k) faults in the n-cube. In this paper several lower and upper bounds for f(n, k) are derived such that the resulting gaps are "small". For instance if k ≥ 2 is constant, then f(n, k) = θ(log n). Especially for k = 2 and large n: f(n, 2) ∈ [⌈αn⌉ : ⌈αn⌉ + 2] where αn = log n + 1/2 log log n + 1/2. Or if k = ω(log log n) then 2k ≪ f(n, k) ≪ 2(1+ε)k, with ε chosen arbitrarily small. The above upper bounds are obtained by analysing the behaviour of an adversary, who makes "worst-case" distributions of a given number of faulty processors. For k = 2 the distribution is obtained constructively, whereas in the general case only the existence is shown using probabilistic arguments. The above bounds change if the notions are relativized with respect to some given parallel faultchecking procedure P. In this case only those subcubes must be made faulty by the adversary, which are possible outputs of P. In the case k = 2 the notion of directed chromatic index is defined to analyse this situation. Relations between the directed chromatic index and the chromatic number are derived, which are of interest in their own right.