Vertex-disjoint paths joining adjacent vertices in faulty hypercubes (original) (raw)

2019, Theoretical Computer Science

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 |F e | + |F v | ≤ 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 ≤ (P [a, b]) + (P [c, d]) ≤ 2 n − 2|F v | − 2, where 2|((P [a, b]) + (P [c, d])), (a, b), (c, d) ∈ 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 |F e | ≤ 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|F v | vertices respectively in Q n when S = {a, c}, T = {b, d}, and (a, b), (c, d) ∈ E(Q n). Our result is optimal with respect to the number of fault-tolerant elements.