Hamiltonian Laceability of Faulty Hypercubes

Abstract

We consider the occurrence of the combination of edge faults and vertex faults in hypercubes. To preserve the equitability of Qn, we restrict the faults on vertex occurring only on disjoint adjacent pairs. Let F be a subset of V(Qn) ∪ E(Qn) such that F can be decomposed into two parts Fav and Fe where Fav is a union of fav disjoint adjacent pairs of Qn and Fe consists of fe edges of Qn. We show that the Qn - F is still hamiltonian if fav + fe ≤ n - 2. In addition, we prove that there exists a hamiltonian path in Qn - {v} - F between any two vertices in the partite set not containing v if fav + fe ≤ n - 3. Furthermore, all bounds are tight. This generalizes a recent result of Fu (2006).

Edge, vertex and mixed fault diameters

“…Both failures of nodes and failures of connections between them happen and it is desirable that a network is robust in the sense that a limited number of failures does not break down the whole system. A lot of work has been done on various aspects of network fault tolerance, see for example the survey [6] and more recent papers [9,13,15]. In particular the fault diameter with faulty vertices which was first studied in [11] and the edge fault diameter has been determined for many important networks recently [8,7,12,14].…”

“…In most papers either only edge faults or only vertex faults are considered, while the case when both edges and vertices may be faulty is studied rarely. For example [9,13] consider Hamiltonian properties assuming a combination of vertex and edge faults. In our recent work on fault diameter of Cartesian graph products and bundles [3][4][5]2], analogous results were found for both fault diameter and edge fault diameter.…”

Survey on path and cycle embedding in some networks

“…Sun et al [133] showed that Q n −F is hyper hamiltonian if |F | = f av +f e n − 3 for n 3, where f av is the number of disjoint pairs of adjacent vertices in Q n . Hsieh [65] has improved the result of Sun et al by showing that there exists a fault-free cycle of length at least 2 n − 2f v in Q n if f e n − 2 and f e + f v 2n − 4 for n 3.…”

Mixed fault diameter of Cartesian graph bundles II

“…Both failures of nodes and failures of connections between them happen and it is desirable that a network is robust in the sense that a limited number of failures does not break down the whole system. A lot of work has been done on various aspects of network fault tolerance, see for example the survey [8] and the more recent papers [16,24,26]. In particular the fault diameter with faulty vertices, which was first studied in [18], and the edge fault diameter have been determined for many important networks recently [2,3,4,5,10,11,19,25].…”

“…Usually either only edge faults or only vertex faults are considered, while the case when both edges and vertices may be faulty is studied rarely. For example, [16,24] consider Hamiltonian properties assuming a combination of vertex and edge faults. In recent work on fault diameter of Cartesian graph products and bundles [2,3,4,5], analogous results were found for both fault diameter and edge fault diameter.…”