Vertex-disjoint paths joining adjacent vertices in faulty hypercubes

Let Q(n) denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in Q(n) be denoted by F-e and F-v, respectively. In this paper, we investigate Q(n) (n >= 3) with vertical bar F-e vertical bar + vertical bar F-v vertical bar <= n - 3 faulty elements, and demonstrate that there are two fault-free vertex-disjoint paths P[a, b] and P[c, d] satisfying that 2 <= l(P[a,b[) + l(P[c, d]) <= 2(n) - 2 vertical bar F-v vertical bar - 2, where 2 vertical bar(l(P[a, b]) + l(P[c. d])), (a, b), (c, d) is an element of E(Q(n)). The contribution of this paper is: (1) we can quickly obtain the interesting result that Q(n) - F-e is bipancyclic, where vertical bar F-e vertical bar n - 2 and n >= 3; (2) this result is a complement to Chen's part result (Chen (2009) [2]) in that our result shows that there are all kinds of two disjoint-free (S, T)-paths which contain 4, 6. 8, ..., 2(n) - 2 vertical bar F-v vertical bar vertices respectively in Q(n) when S = {a, c}, T = {b, d}, and (a, b). (c, d) is an element of E(Q(n)). Our result is optimal with respect to the number of fault-tolerant elements. (C) 2019 Elsevier B.V. All rights reserved.