The bipancycle-connectivity of the hypercube

Abstract

Paths and cycles are popular interconnection networks due to their simplicity and low degrees, and therefore an efficient way of embedding them into an interconnection network is of particular importance. Many path-embedding aspects and cycle-embedding aspects, such as Hamiltonicity, Hamiltonian-connectivity, panconnectivity, bipanconnectivity, pancyclicity, vertex-pancyclicity and edge-pancyclicity, have been proposed and extensively studied. In fact, these discussions are meaningful for interconnection networks of resource-allocated systems or heterogeneous computing systems. In this paper, we propose a new cycle-embedding aspect called bipancycle-connectivity that combines the concepts of vertex-pancyclicity and bipanconnectivity. A bipartite graph is bipancycle-connected if an arbitrary pair of vertices x, y is contained by common cycles called bipanconnected cycles that include a cycle of every even length ranging from minc( x, y) to N; where minc( x, y) denotes the length of the minimum cycle that contains x and y, and N is the number of vertices. In this paper, we introduce a new graph called the cycle-of-ladders. We show that the hypercube is bipancycle-connected by presenting algorithms to embed the cycle-of-ladders into the hypercube.