Conditional fault-tolerant cycles in folded hypercubes with faulty elements

Abstract

ABSTRACT As an attractive variation of the hypercubes Q n , n -dimensional folded hypercube F Q n can be obtained by adding 2 n1 complementary edges between the vertices of hypercube. Let F F v (respectively,F F e ) denotes the set of faulty vertices (respectively, faulty edges) in an n -dimensional folded hypercube. In this paper, we prove that F QFF FF nve  contains a fault-free cycle of length at least 2 n  F Fv if F Q n satisfies both of the constraints that A1 Aeach vertex in F Q n is incident to at least two fault-free edges A2 A FF FF n ev  d23, when n t 4 . Keywords: Folded hypercubes, faulty elements , fault tolerance, cycle embedding. 1. INTRODUCTION As an interconnection network topology in multi-processor system, hypercubes structure has been a household word in practical applications for its attractive properties [1]. On e of the variations of hypercubes proposed by El-Amawy and Latifi [2] is the folded hypercube which can be established by adding an edge to every pair of vertices with complementary addresses. Apart from the folded hypercube, there are many variations of the hypercube being put forward [3,4], which makes the portability of an algorithm across these networks to be considerable importance. A guest nework G can be embedded into a host network H means that the algorithms on the guest can be simulated on the host systematically. Embedding is a form of a one-to one mapping f , from the vertices set in G into that in H so that an edge of G corresponds to a path of H under f . Paths and cycles are two of the popular networks for parallel computation E research men are obsessed with the embedding of them into different networks [5-11]. However, vertices or edges faults may develop when the network is in use, hence, it makes the study on faulty networks to be practical significance. Two models proposed by the scholars are used to embed fault-tolerant cycles. One is the standard fault model, which assumes that the distribution of the faulty elements are free to any restrictions, the other is the conditional fault model, which assumes that each ve rtex of the faulty networks is connected to at least two fault-free edges. Results of the folded hypercube in this research field is quite a lot [6-10], for instance.Fu [7] solved the problem of how to embed a fault-free cycle of length at least 22