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,…,k_n)$T=T(k_1,k_2,\dots ,k_n)$ be a torus and F be a set of faulty edges in T.

Let |F|≤2n−3$|F|\le 2n-3$ and even k_i≥4$k_i\ge 4$ for all i  . Then T−F$T-F$ has a paired 2-DPC if each partite set contains two end-vertices.

Let 132b309a" title="Click to view the MathML source">|F|≤2n−4$|F|\le 2n-4$ and odd k_i≥3$k_i\ge 3$ for all i   with at most one even k_j$k_j$. Then T−F$T-F$ has a paired 2-DPC.

Our brief proofs are based on a technique that is of interest and may find some applications.