The Bipancycle-Connectivity and the m-Pancycle-Connectivity of the k-ary n-cube
J.-F. Fang · The Computer Journal · 2009
The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks that are desirable in both theoretical interests and practical systems, including the ring, torus and hypercube, may be regarded as the subclasses of k-ary n-cubes. In this paper, we investigate the pancycle-connected properties of the k-ary n-cube. We show that the k-ary n-cube is bipancycle-connected for k being even. That is, each pair of vertices x and y is contained by a cycle of each even length ranging from the length of the smallest even cycle that contains x and y to N, where N is the order of the network. We also show that the k-ary n-cube is strictly m-pancycle-connected for k being odd and n ≥ 2, where m = nk - n. That is, each pair of vertices is contained by a cycle of each length ranging from nk - n to N; and nk - n has reached the lower bound of the problem.