On embedding cycles into faulty dual-cubes

A dual-cube uses low-dimensional hypercubes as basic components such that keeps the main desired properties of the hypercube. Each hypercube component is referred as a cluster. A (n+1)-connected dual-cube DC(n) has 22n+1 nodes and the number of nodes in a cluster is n2. There are two classes with each class consisting of n2 clusters. Each node is incident with exactly n+1 links where n is the degree of a cluster, one more link is used for connecting to a node in another cluster. In this paper, we show that every node of DC(n) lies on a cycle of every even length from 4 to 22n+1 inclusive for n⩾3, that is, DC(n) is node-bipancyclic for n⩾3. Furthermore, we show that DC(n), n⩾3, is bipancyclic even if it has up to n−1 edge faults. The result is optimal with respect to the number of edge faults tolerant.