Paired 2-disjoint path covers of multidimensional torus networks with faulty edges

A paired k-disjoint path cover (paired k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all vertices of the graph. Clearly, the paired k-DPC is stronger than Hamiltonian-connectivity. The n-dimensional torus T(k1,k2,…,kn) (including the k-ary n-cube Qnk) is one of the most popular interconnection networks. In this paper, we obtain the following results. (1) Assume even ki≥4 for i=1,2,…,n. Let T=T(k1,k2,…,kn) be a bipartite torus and F be a set of faulty edges with |F|≤2n−3. Given any four vertices s1,t1,s2 and t2, such that each partite set contains two vertices. Then the graph T−F has a paired 2-DPC consisting of s1−t1 path and s2−t2 path. And the upper bound 2n−3 of edge faults tolerated is optimal. The result is a generalization of the result of Park et al. concerning the case of n=2[17]. (2) Assume ki≥3 for i=1,2,…,n, with at most one ki being even. Let T=T(k1,k2,…,kn) be a torus and F be a set of faulty edges with |F|≤2n−4. Then the graph T−F has a paired 2-DPC. And the upper bound 2n−4 of edge faults tolerated is nearly optimal. The result is a generalization of the result of Park concerning the case of n=2[16]. Our brief proofs are based on a technique that is of interest and may find some applications. •A paired k-DPC of a graph is a set of k disjoint paths joining k source-sink pairs that cover all vertices of the graph.•Let T=T(k1,k2,…,kn) be a torus and F be a set of faulty edges in T.•Let |F|≤2n−3 and even ki≥4 for all i. Then T−F has a paired 2-DPC if each partite set contains two end-vertices.•Let |F|≤2n−4 and odd ki≥3 for all i with at most one even kj. Then T−F has a paired 2-DPC.•Our brief proofs are based on a technique that is of interest and may find some applications.