The panconnectivity and the pancycle-connectivity of the generalized base-b hypercube

The interconnection network considered in this paper is the generalized base-b hypercube that is an attractive variant of the well-known hypercube. The generalized base-b hypercube is superior to the hypercube in many criteria, such as diameter, connectivity, and fault diameter. In this paper, we study the panconnectivity and pancycle-connectivity of the generalized base-b hypercube. We show that a generalized base-b hypercube is panconnected for b≥3. That is, for each pair of distinct vertices x and y of the n-dimensional generalized base-b hypercube GH(b,n) and for any integer l, where Dist(x,y)≤l≤N−1, there exists a path of the length l joining x and y, where N is the order of the graph GH(b,n) and Dist(x,y) is the distance between x and y. We also show that a generalized base-b hypercube is pancycle-connected for b≥3. That is, every two distinct vertices x and y of the graph GH(b,n) are contained by a cycle of every length ranging from the length of the smallest cycle that contains x and y to N.