Paired 2-disjoint path covers of Bcube under the partitioned edge fault model

BCube, as a popular server-centric data center network (DCN), offers significant advantages in low latency, load balancing, and high bandwidth. The many-to-many paired $m$-disjoint path cover ($m$-DPC), a generalization of Hamiltonian paths, enhances message transmission efficiency by constructing disjoint paths that connect $m$ source-destination pairs while covering all the nodes. However, with the continuous expansion of DCNs, link and service failures have grown increasingly common, necessitating robust fault-tolerant algorithms to guarantee reliable communication.This paper mainly investigates the fault-tolerant paired 2-DPC embedding in BCube. We prove that under the partitioned edge fault (PEF) model, BCube retains a paired 2-DPC even when exponentially many edge failures occur.

💡 Research Summary

The paper investigates the fault‑tolerant embedding of paired two‑disjoint‑path covers (paired 2‑DPC) in the server‑centric data‑center network BCube. A paired 2‑DPC consists of two vertex‑disjoint paths that connect two source‑destination pairs while collectively visiting every node of the underlying graph. The authors adopt the partitioned edge‑fault (PEF) model, which limits the number of faulty edges in each dimension of the BCube topology, reflecting realistic failure scenarios where a node does not lose all incident links simultaneously.

After reviewing related work on Hamiltonian connectivity and m‑disjoint‑path covers (m‑DPC) in various interconnection networks, the authors formalize the PEF model for BCube(n, k). For each dimension i (0 ≤ i ≤ k) they define an upper bound f(i) on the number of faulty edges allowed in that dimension:

• For the 0‑th dimension (internal edges of the (k‑1)‑dimensional sub‑cubes) f(0) ≤ n − 5 when n = 7, otherwise f(0) ≤ n − 4.
• For dimensions 1…k, when 4 ≤ n ≤ 9, f(i) ≤ max{0, n_i − 5}; when n ≥ 10, f(i) ≤ ⌈(n_i − 1)/2⌉·(n − 2) − 3n/2.

Using known results that a complete graph K_n remains Hamiltonian‑connected after up to (n − 4) edge deletions, the authors first prove that K_n − F is paired 2‑DPC‑coverable under the same fault bound (Claim 1). They then extend this property to BCube by exploiting its recursive structure: each (k‑1)‑dimensional sub‑cube B C