Geodesic pancyclicity and balanced pancyclicity of the generalized base-b hypercube

Recently, Chan et al. have proposed a new cycle embedding property called balanced pancyclicity [Discrete Applied Mathematics 155(15) (2007) 1971-1978]. For a graph G(V, E) and any two nodes x and y of V, a cycle R contains x and y can be divided into two paths, Pt1 and Pt2, joining x and y such that len(Pt1)len(Pt2), where len() denote the length of the path . A balanced cycle of an even (respectively, odd) length, len(Pt1) = len(Pt2) (respectively, len(Pt1) = len(Pt2)-1). A graph is balanced pancyclic if every two nodes x and y are contained in a balanced cycle from Max(3, 2Dist(x, y)) to N, where N is the order of the graph. The interconnection network considered in this paper is the generalized base-b hypercube that is an attractive variant of the well-known hypercube. In fact, the generalized base-b hypercube is the Cartesian products of complete graphs with b nodes. 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 balanced pancyclicity of the generalized base-b hypercube. We show that the generalized base-b hypercube is balanced pancyclic for b4.