Fault-Tolerant Hamiltonicity and Hamiltonian Connectivity of BCube with Various Faulty Elements

Abstract

BCube is one kind of important data center networks. Hamiltonicity and Hamiltonian connectivity have significant applications in communication networks. So far, there have been many results concerning fault-tolerant Hamiltonicity and fault-tolerant Hamiltonian connectivity in some data center networks. However, these results only consider faulty edges and faulty servers. In this paper, we study the fault-tolerant Hamiltonicity and the fault-tolerant Hamiltonian connectivity of BCube(n, k) under considering faulty servers, faulty links/edges, and faulty switches. For any integers n ≥ 2 and k ≥ 0, let BCn,k be the logic structure of BCube(n, k) and F be the union of faulty elements of BCn,k. Let fv, fe, and fs be the number of faulty servers, faulty edges, and faulty switches of BCube(n, k), respectively. We show that BCn,k − F is fault-tolerant Hamiltonian if fv +fe + (n − 1)fs ≤ (n − 1)(k + 1) − 2 and BCn,k −F is fault-tolerant Hamiltonian-connected if fv + fe + (n − 1)fs ≤ (n − 1)(k + 1) − 3. To the best of our knowledge, this paper is the first work which takes faulty switches into account to study the fault-tolerant Hamiltonicity and the fault-tolerant Hamiltonian connectivity in data center networks.