Hyper-Hamiltonian laceability of balanced hypercubes

Abstract

The balanced hypercube, proposed by Wu and Huang, is a new variation of hypercube. The particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. A Hamiltonian bipartite graph with bipartition $$V_{0}\cup V_{1}$$ V 0 Â? V 1 is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $$x\in V_{0}$$ x Â? V 0 and $$y\in V_{1}$$ y Â? V 1 . A graph $$G$$ G is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $$v\in V_{i}$$ v Â? V i , $$i\in \{0,1\}$$ i Â? { 0 , 1 } , there is a Hamiltonian path in G---v between any pair of vertices in $$V_{1-i}$$ V 1 - i . In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.