Embedding cycles and paths in a k-ary n-cube

The k-ary n-cube, denoted by Qn^k, has been one of the most common interconnection networks. In this paper, we study some topological properties of Qn^k. Given two arbitrary distinct nodes x and y in Qn^k, we show that there exists an x–y path of every length from [k/2]n to kⁿ − 1, where n ≥ 2 is an integer and k ≥ 3 is an odd integer. Based on this result, we further show that each edge in Qn^k lies on a cycle of every length from k to kⁿ. In addition, we show that Qn^k is both bipanconnected and edge-bipancyclic, where n ≥ 2 is an integer and k ≥ 2 is an even integer.