Paired 2-disjoint path covers of faulty k-ary n-cubes

A paired k-disjoint path cover of a graph is a set of k disjoint paths joining k distinct source-sink pairs that cover all vertices of the graph. The k-ary n-cube Qnk is one of the most popular interconnection networks. In this paper, we consider the problem of paired 2-disjoint path covers of the k-ary n-cube Qnk (odd k≥5) with faulty elements (vertices and/or edges) and obtain the following result. Let F be a set of faulty elements in Qnk (odd k≥5) with |F|≤2n−4, and {a,b} and {c,d} be its any two pairs of non-faulty vertices. Then the graph Qnk−F contains vertex-disjoint a−b path and c−d path that cover its all non-faulty vertices, and the upper bound 2n−4 of faults tolerated is nearly optimal. We also show that Cartesian product Qnk×P with at most 2n−2 faulty elements is Hamiltonian connected, where P is a path, n≥2 and odd k≥5.