On the construction of paired many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements

A paired many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. This paper is concerned with paired many-to-many disjoint path coverability of hypercube-like interconnection networks, called restricted HL-graphs. The class includes twisted cubes, crossed cubes, multiply twisted cubes, Möbius cubes, Mcubes, and generalized twisted cubes. We show that every restricted HL-graph of degree m with f or less faulty elements has a paired many-to-many k-DPC for any f and k ≥ 2 with f + 2k ≤ m. The result improves the known bound of f + 2k ≤ m − 1 by one.