Disjoint path covers with path length constraints in restricted hypercube-like graphs.

A disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. In this paper, we prove that given k sources, s1, …, sk, in an m-dimensional restricted hypercube-like graph with a set F of faults (vertices and/or edges), associated with k positive integers, l1, …, lk, whose sum is equal to the number of fault-free vertices, there exists a disjoint path cover composed of k fault-free paths, each of whose paths starts at si and contains li vertices for i∈{1,…,k}, provided |F|+k≤m−1. The bound, m−1, on |F|+k is the best possible.