Fault-tolerant embedding of cycles of various lengths in k-ary n-cubes

The k-ary n-cube has been one of the most popular interconnection networks for distributed-memory parallel systems. In this paper, we study the problem of embedding cycles of various lengths into faulty k-ary n-cubes. It is proved that a faulty k-ary n  -cube with fvfv faulty vertices and fefe faulty edges admits a fault-free cycle of every even length from 4 to kn−2fvkn−2fv if k⩾4k⩾4 is even and fv+fe⩽2n−2fv+fe⩽2n−2. Furthermore, we show that every healthy edge of the faulty k-ary n  -cube lies on a fault-free cycle of every even length from 4 to kn−2fvkn−2fv if k⩾4k⩾4 is even and fv+fe⩽2n−3fv+fe⩽2n−3. The results are optimal with respect to the number of vertex and edge faults tolerated.