Path Embeddings with Prescribed Edge in the Balanced Hypercube Network

Abstract: The balanced hypercube network, which is a novel interconnection network for parallel computation and data processing, is a newly-invented variant of the hypercube. The particular feature of the balanced hypercube is that each processor has its own backup processor and they are connected to the same neighbors. A Hamiltonian bipartite graph with bipartition V0 ∪V1 is Hamiltonian laceable if there exists a path between any two vertices x ∈ V0 and y ∈ V1. It is known that each edge is on a Hamiltonian cycle of the balanced hypercube. In this paper, we prove that, for an arbitrary edge e in the balanced hypercube, there exists a Hamiltonian path between any two vertices x and y in different partite sets passing through e with e 6= xy. This result improves some known results.