Embedding various cycles with prescribed paths into k-ary n-cubes

The k-ary n-cube has been one of the most popular interconnection networks for large-scale multi-processor systems and data centers. In this study, we investigate the problem of embedding cycles of various lengths passing through prescribed paths in the k-ary n-cube. For n≥2 and k≥5 with k odd, we prove that every path with length h (1≤h≤2n−1) in the k-ary n-cube lies on cycles of every length from h+(k−1)(n−1)/2+k to kn inclusive.