## 1 Introduction  
The data center network (DCN) is a critical infrastructure of cloud computing, which can provide important cloud services such as GFS [1], Bigtable [2], and Dryad [3]. A data center network is a bridge connecting large-scale servers in the data center for distributed computing. As a widely used server-centric DCN, BCCC [4] can provide good network performance using inexpensive off-the-shelf switches and commodity servers with only two network interface card (NIC) ports. BCCC has many desirable properties such as high scalability, a near-equal-length parallel path, and a small diameter. The Hamilton properties and routing algorithm of BCCC have been studied [5; 6; 7; 8]. Server failures are normal, since there are a large number of servers in the actual network. When the network fails, how to restore normal communication should be considered first. Furthermore, whether there is a fault-free path between any two different fault-free vertices and how to construct a fault-free path using an algorithm should be also taken into account. In other words, this is a problem of fault-tolerant routing, which is also the major problem we will study in this paper. When designing or selecting a network topology for a data center, the most fundamental consideration is the fault tolerance of the network. Fault tolerance refers to the ability to continue working in the case of a network failure. Fault tolerance is the maximum number of failed devices that the network will allow to operate normally. In this paper, fault tolerance is directly determined by the connectivity of the network. The larger the connectivity scale, the better fault-tolerant performance of the data center network. In order to overcome the shortcoming of traditional connectivity, [PERSON] [9] introduced the concept of the restricted faulty nodes on the resulting graph. Following this trend, [PERSON] [10] proposed the concept of restricted connectivity. The restricted connectivity is the connectivity under the condition that each fault-free node has at least one fault-free neighbor. There have been many studies on the restricted connectivity of networks. [PERSON] et al. [11] establish a universally \(h\)-restricted connectivity for a class of hypercube-based compound networks, such as hierarchical cubic network and its generalization complete cubic network. [PERSON] [12] studies the \(h\)-restricted connectivity of an n-dimensional balanced hypercube \(BH_{n}\). [PERSON] et al. [13] investigate the h-restricted connectivity of the generalized hypercube. [PERSON] et al. [14] prove the k-restricted edge connectivity of the data center network DCell and [PERSON] et al. [15] establish the 3-restricted connectivity of an n-dimensional split-star network. [PERSON] [16] shows the r-restricted connectivity of hyper Petersen graphs. [PERSON] et al. [17] prove the restricted edge connectivity of Kronecker product graphs. Finding disjoint paths between two vertex sets in a massively parallel system establishes secure communication paths that do not interfere with each other. Constructing multiple disjoint paths for two vertices increases the probability of constructing a fault-free path. In addition to being used to avoid congestion, speed up transmission rates, and provide alternative propagation paths, disjoint paths can also enhance the robustness of vertex failures and the ability to load balance. [PERSON] et al. [18] propose an algorithm to give the disjoint path between any two distinct vertices in dragonfly networks. [PERSON] [19] gives exact algorithms for disjoint paths and disjoint connected subgraphs on graphs with n vertices and m edges in H-free graphs. [PERSON] et al. [20] propose a self-stabilizing One-to-Many node disjoint path routing algorithm in star networks. Fault-tolerant techniques are one of the pillar technologies of the network. At the routing algorithm level, routing fault tolerance is achieved by selecting different routing paths to bypass faulty links and nodes.