Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes

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Highlights

  • Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes was investigated.

  • Fault-free hamiltonian paths passing through prescribed linear forests in faulty balanced hypercubes were constructed.

  • Some known results were improved.

Abstract

Given a bipartite graph G with bipartition (X,Y), let FE(G) be a set of faulty edges and let L be a linear forest in GF such that |F|+|E(L)|k. Let uX and vY be any two vertices such that none of the paths in L has u or v as internal vertices or both of them as end vertices. G is said to be k-fault-tolerant-prescribed hamiltonian laceable if GF admits a hamiltonian path between u and v passing through L. Balanced hypercubes are candidate interconnection networks of multiprocessor systems. In this paper, we prove that the n-dimensional balanced hypercube is (n1)-fault-tolerant-prescribed hamiltonian laceable.

Section snippets

Introduction and preliminaries

For the graph-theoretical terminology, notation and operation used without defining here we follow [1] and [9]. Set Nn={0,1,,n1} for a positive integer n.

Let G be a bipartite graph with bipartition (X,Y), and let u and v be two arbitrary vertices of X and Y, respectively. We refer to a path of G between u and v as π[u,v], and refer to a “hamiltonian path” as an “H-path”. G is hamiltonian laceable if G has an H-path π[u,v] [6]. Let FE(G) be an arbitrary set of at most f faulty edges. G is

Proof of Theorem 1.6

In this section, all additions and subtractions on some digit of a vertex in BHn are modulo 4 unless otherwise stated.

Let FE(BHn) be a set of faulty edges and let L be linear forest in BHnF such that |F|+|E(L)|n1. Let uX and vY such that {u,v} and L are compatible. In the remainder of the section, we will prove that BHn admits a fault free H-path π[u,v] passing through L. It suffices to show that BHnF has an H-path π[u,v] for the case that |E(L)|+|F| reaches its upper bound n1.

By

Concluding remarks

In this paper, we proved that the n-dimensional balanced hypercube BHn is (n1)-fault-tolerant-prescribed hamiltonian laceable for n1, which extended Chen et al.'s result [2] and Lü et al.'s result [5]. Note that BH1 is a rectangle on the four vertices labelled by 0, 1, 2 and 3. Clearly, BH1(0,1) has no H-path between the vertices 1 and 2, which implies that the upper bound (n1) on |F|+|E(L)| cannot be improved for n=1. The problem of the fault-tolerant-prescribed hamiltonian laceability of B

Acknowledgement

The authors would like to express their heartfelt thanks to the referees for their comments and suggestions which are helpful to improve the quality of this paper. This work is partly supported by the Joint Fund of NSFC and Henan Province (U1304601).

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