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 $V_{0}\cup V_{1}$ is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $x\in V_{0}$ and $y\in V_{1}$. A graph $G$ is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $v\in V_{i}$, $i\in \{0,1\}$, there is a Hamiltonian path in G–v between any pair of vertices in $V_{1-i}$. In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.