# Embedding Hamiltonian Cycles into Folded Hypercubes with Faulty Links

Research output: Contribution to journalArticle

70 Citations (Scopus)

### Abstract

It has been known that an n-dimensional hypercube (n-cube for short) can always embed a Hamiltonian cycle when the n-cube has no more than n-2 faulty links. In this paper, we study the link-fault tolerant embedding of a Hamiltonian cycle into the folded hypercube, which is a variant of the hypercube, obtained by adding a link to every pair of nodes with complementary addresses. We will show that a folded n-cube can tolerate up to n-1 faulty links when embedding a Hamiltonian cycle. We present an algorithm, FT_HAMIL, that finds a Hamiltonian cycle while avoiding any set of faulty links F provided that F≤n-1. An operation, called bit-flip, on links of hyper-cube is introduced. Simple yet elegant, bit-flip will be employed by FT_HAMIL as a basic operation to generate a new Hamiltonian cycle from an old one (that contains faulty links). It is worth pointing out that the algorithm is optimal in the sense that for a folded n-cube, n-1 is the maximum number for F that can be tolerated, F being an arbitrary set of faulty links.

Original language English 545-564 20 Journal of Parallel and Distributed Computing 61 4 https://doi.org/10.1006/jpdc.2000.1681 Published - 1 Apr 2001

### Fingerprint

Hamiltonians
Hamiltonian circuit
N-cube
Hypercube
Flip
Fault-tolerant Embedding
Regular hexahedron
n-dimensional
Arbitrary
Vertex of a graph

### Keywords

• Embedding; fault tolerance; folded hypercube; Hamiltonian cycle; link fault

### Cite this

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title = "Embedding Hamiltonian Cycles into Folded Hypercubes with Faulty Links",
abstract = "It has been known that an n-dimensional hypercube (n-cube for short) can always embed a Hamiltonian cycle when the n-cube has no more than n-2 faulty links. In this paper, we study the link-fault tolerant embedding of a Hamiltonian cycle into the folded hypercube, which is a variant of the hypercube, obtained by adding a link to every pair of nodes with complementary addresses. We will show that a folded n-cube can tolerate up to n-1 faulty links when embedding a Hamiltonian cycle. We present an algorithm, FT_HAMIL, that finds a Hamiltonian cycle while avoiding any set of faulty links F provided that F≤n-1. An operation, called bit-flip, on links of hyper-cube is introduced. Simple yet elegant, bit-flip will be employed by FT_HAMIL as a basic operation to generate a new Hamiltonian cycle from an old one (that contains faulty links). It is worth pointing out that the algorithm is optimal in the sense that for a folded n-cube, n-1 is the maximum number for F that can be tolerated, F being an arbitrary set of faulty links.",
keywords = "Embedding; fault tolerance; folded hypercube; Hamiltonian cycle; link fault",
author = "Dajin Wang",
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In: Journal of Parallel and Distributed Computing, Vol. 61, No. 4, 01.04.2001, p. 545-564.

Research output: Contribution to journalArticle

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AU - Wang, Dajin

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AB - It has been known that an n-dimensional hypercube (n-cube for short) can always embed a Hamiltonian cycle when the n-cube has no more than n-2 faulty links. In this paper, we study the link-fault tolerant embedding of a Hamiltonian cycle into the folded hypercube, which is a variant of the hypercube, obtained by adding a link to every pair of nodes with complementary addresses. We will show that a folded n-cube can tolerate up to n-1 faulty links when embedding a Hamiltonian cycle. We present an algorithm, FT_HAMIL, that finds a Hamiltonian cycle while avoiding any set of faulty links F provided that F≤n-1. An operation, called bit-flip, on links of hyper-cube is introduced. Simple yet elegant, bit-flip will be employed by FT_HAMIL as a basic operation to generate a new Hamiltonian cycle from an old one (that contains faulty links). It is worth pointing out that the algorithm is optimal in the sense that for a folded n-cube, n-1 is the maximum number for F that can be tolerated, F being an arbitrary set of faulty links.

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