Distributed algorithms for building Hamiltonian cycles in k-ary n-cubes and hypercubes with faulty links

We derive a sequential algorithm Find-Ham-Cycle with the following property. On input: k and n (specifying the k-ary n-cube Q_n ^k); F, a set of at most 2n - 2 faulty links; and v, a node of Q_n ^k, the algorithm outputs nodes v⁺ and v^- such that if Find-Ham-Cycle is executed once for every node v of Q_n ^k then the node v⁺ (resp. v ^-) denotes the successor (resp. predecessor) node of v on a fixed Hamiltonian cycle in Q_n ^k in which no link is in F. Moreover, the algorithm Find-Ham-Cycle runs in time polynomial in n and log k. We also obtain a similar algorithm for an n-dimensional hypercube with at most n - 2 faulty links. We use our algorithms to obtain distributed algorithms to embed Hamiltonian cycles in k-ary n-cubes and hypercubes with faulty links