Vertex-disjoint paths joining adjacent vertices in faulty hypercubes

Abstract

Let Qn denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in Qn be denoted by Fe and Fv, respectively. In this paper, we investigate Qn (n≥3) with |Fe|+|Fv|≤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])≤2n−2|Fv|−2, where 2|(ℓ(P[a,b])+ℓ(P[c,d])), (a,b),(c,d)∈E(Qn). The contribution of this paper is: (1) we can quickly obtain the interesting result that Qn−Fe is bipancyclic, where |Fe|≤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,…,2n−2|Fv| vertices respectively in Qn when S={a,c}, T={b,d}, and (a,b),(c,d)∈E(Qn). Our result is optimal with respect to the number of fault-tolerant elements.