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.

中文翻译：

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BCube的容错哈密顿性和哈密顿连通性

BCube 是一种重要的数据中心网络。哈密​​顿性和哈密顿连通性在通信网络中具有重要的应用。到目前为止，在一些数据中心网络中已经有很多关于容错哈密顿性和容错哈密顿连通性的结果。然而，这些结果只考虑了故障边缘和故障服务器。在本文中，我们研究了在考虑故障服务器、故障链路/边缘和故障交换机的情况下 BCube(n, k) 的容错哈密顿性和容错哈密顿连通性。对于任意整数n≥2和k≥0，设BCn,k为BCube(n,k)的逻辑结构，F为BCn,k的错误元素的并集。设 fv、fe 和 fs 分别是 BCube(n,k) 的故障服务器、故障边和故障交换机的数量。我们证明BCn，