Fault-tolerant cycle embedding in dual-cube with node faults

Abstract

A low-degree dual-cube was proposed as an alternative to the hypercubes. A dual-cube DC(m) has m + 1 links per node where m is the degree of a cluster (m-cube) and one more link is used for connecting to a node in another cluster. There are 2 m+1 clusters and hence the total number of nodes is 2 2m+1 in a DC(m). In this paper, by using Gray code, we show that there exists a faulty-free cycle containing at least 2 2m+1 − 2 f nodes with f ≤ m − 1 faulty nodes.

Mutually independent Hamiltonian cycles in dual-cubes

“…Its nice structure has drawn the attention of many researchers [11,[15][16][17][18][19][20]. A dual-cube DC n is obtained from a basic component Q n as follows.…”

“…By the definition of DC n and the study of [20], we know that DC n is an (n + 1)-regular bipartite graph. Any vertex u in DC n is adjacent to n vertices in the same cluster and to one vertex in some cluster of the other class.…”

“…Although any DC n has much less edges than Q 2n+1 with the same number of vertices, the diameter of DC n , 2n + 2, is of the same order of the diameter of Q 2n+1 , which is 2n + 1. Other advanced subjects such as fault-tolerant cycle embedding and multiple disjoint paths construction in dual-cubes are also investigated [13,[15][16][17][18][19][20].…”

Conditional edge-fault-tolerant Hamiltonicity of dual-cubes

“…Accordingly, various fault-tolerant measures have been proposed in the literature, including fault diameter, fault Hamiltonicity, fault pancyclicity, fault Hamiltonian laceability, etc. The random fault model considered in [3,4,[8][9][10]12,14,16,23], assumed faults might happen anywhere in a network without any restriction. On the other hand, under the assumption each node is incident with two or more fault-free links, it was shown [1,2] a k-ary, k P 3, n-dimensional hypercube (an n-dimensional hypercube) contains a fault-free Hamiltonian cycle, even if there are up to 4n À 5 (2n À 5) link faults.…”

On embedding cycles into faulty dual-cubes

“…Li et al [4] showed that DC(n) contains a fault-free Hamiltonian cycle even if it has up to n − 1 edge faults for n 2. Subsequently, they showed [5] that there exists a fault-free cycle containing at least 2 2n+1 − 2 f vertices in DC(n), n 3, with f n faulty nodes.…”