Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links

The k-ary n-cube, denoted by Q_n^k, is one of the most important interconnection networks for parallel computing. In this paper, we consider the problem of embedding cycles and paths into faulty 3-ary n-cubes. Let F be a set of faulty nodes and/or edges, and n ≥ 2. We show that when | F | ≤ 2 n - 2, there exists a cycle of any length from 3 to | V (Q_n³ - F) | in Q_n³ - F. We also prove that when | F | ≤ 2 n - 3, there exists a path of any length from 2 n - 1 to | V (Q_n³ - F) | - 1 between two arbitrary nodes in Q_n³ - F. Since the k-ary n-cube is regular of degree 2 n, the fault-tolerant degrees 2 n - 2 and 2 n - 3 are optimal.