One-to-Many Disjoint Path Covers in a Graph with Faulty Elements

In a graph G, k disjoint paths joining a single source and k distinct sinks that cover all the vertices in the graph are called a one-tomany k-disjoint path cover of G. We consider a k-disjoint path cover in a graph with faulty vertices and/or edges obtained by merging two graphs H0 and H1, |V (H0)| = |V (H1)| = n, with n pairwise nonadjacent edges joining vertices in H0 and vertices in H1. We present a sufficient condition for such a graph to have a k-disjoint path cover and give the construction scheme. Applying our main result to interconnection graphs, we observe that when there are f or less faulty elements, all of recursive circulant G(2, 4), twisted cube TQm, and crossed cube CQm of degree m have k-disjoint path covers for any f ≥ 0 and k ≥ 2 such that f + k ≤ m− 1.