Embedding paths of variable lengths into hypercubes with conditional link-faults

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Embedding paths of variable lengths into hypercubes with conditional

link-faults

Tz-Liang Kueng

a

, Cheng-Kuan Lin

b

, Tyne Liang

b,*

, Jimmy J.M. Tan

b

, Lih-Hsing Hsu

c,1 a

Department of Computer Science and Information Engineering, Asia University, 500 Lioufeng Rd., Taichung, Taiwan 41354, ROC

b

Department of Computer Science, National Chiao Tung University, 1001 University Rd., Hsinchu, Taiwan 30050, ROC

c

Department of Computer Science and Information Engineering, Providence University, 200 Chung Chi Rd., Taichung, Taiwan 43301, ROC

a r t i c l e

i n f o

Article history:

Received 10 September 2007 Received in revised form 18 September 2008

Accepted 26 June 2009 Available online 2 July 2009 Keywords: Interconnection network Hypercube Fault tolerance Conditional fault Linear array Path embedding

a b s t r a c t

Faults in a network may take various forms such as hardware failures while a node or a link stops functioning, software errors, or even missing of transmitted packets. In this paper, we study the link-fault-tolerant capability of an n-dimensional hypercube (n-cube for short) with respect to path embedding of variable lengths in the range from the shortest to the longest. Let F be a set consisting of faulty links in a wounded n-cube Qn, in which every node is still incident to at least two fault-free links. Then we show that Qn F has a path of any odd (resp. even) length in the range from the distance to 2n_{1 (resp. 2}n₂₎ between two arbitrary nodes even if jFj ¼ 2n 5. In order to tackle this problem, we also investigate the fault diameter of an n-cube with hybrid node and link faults.

1. Introduction

In many parallel computer systems, processors are connected on the basis of interconnection networks. Such networks usually have a regular degree, i.e., every node is incident to the same number of links. Popular instances of interconnection networks include hypercubes, star graphs, meshes, bubble-sort networks, etc.

The hypercube is one of the most versatile interconnection networks yet discovered for parallel computation. It can efﬁ-ciently simulate many other networks of various sizes[14]. Because nodes and/or links in a network may fail accidentally, it is demanded to consider fault tolerance of a network. Hence, the issue of faulty hypercubes has been widely addressed in researches[2,4,11,16,20–24]. For example, Latiﬁ et al. [11]proved that an n-dimensional hypercube (n-cube for short) has a hamiltonian cycle even if it has n 2 faulty links. Furthermore, Li et al.[16]showed that an n-cube is bipancyclic even if it has up to n 2 faulty links; Tsai et al.[20]showed that a faulty n-cube is both hamiltonian laceable and strongly ham-iltonian laceable if it has n 2 faulty links. Recently, Xu et al.[24]showed that an n-cube, with n 2 faulty links, contains a path of length l between any two nodes of distance dfor each integer l satisfying d6_{l 6 2}n_{1 and 2jðl d}_{Þ, where} expres-sion 2jðl dÞ means that l d 0 ðmod 2Þ. Moreover, Fu[4]proved that a fault-free path of length at least 2n 2f 1 (or 2n

2f 2) can be embedded to join two arbitrary nodes of odd (or even) distance in an n-cube with f 6 n 2 faulty nodes. Since linear array and rings are two of the most fundamental structures for parallel and distributed computation, a variety of efﬁcient algorithms were developed on these two topologies[14]. In particular, embedding of linear array and rings in a

*Corresponding author. Tel.: +886 3 5131365; fax: +886 3 5721490. E-mail address:[email protected](T. Liang).

1

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-233-002.

Contents lists available atScienceDirect

Parallel Computing

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faulty interconnection network is of great signiﬁcance. For example, path embedding in a faulty n-cube was addressed in

[16,20,24]. However, one should notice that each component of a network may have different reliability. Thus, the probabil-ity that all faulty components would be close to one another seems low. With this observation, Harary[7]first introduced the concept of conditional connectivity. Later, Latifi et al.[13]defined the conditional node-faults, which require each node of a network to have at least g fault-free neighbors. It is intuitive to extend this concept by defining conditional link-faults, which require that every node will be incident to at least g fault-free links. In this paper, we only concern g ¼ 2. For convenience, we say a network is conditionally faulty if and only if every node is incident to at least two fault-free links. Under this assumption, Chan and Lee[2]discussed the existence of hamiltonian cycles in an n-cube with 2n 5 conditional link-faults. In addition, Tsai[21]showed that an injured n-cube contains a fault-free cycle of every even length from 4 to 2ninclusive even if it has up to 2n 5 conditional link-faults. It was also proved in[21]that an n-cube with 2n 5 conditional link-faults is hamiltonian laceable and strongly hamiltonian laceable.

As Shih et al.[18]showed, any fault-free link of a faulty n-cube lies on a cycle of even length in the range from 6 to 2n

when up to 2n 5 conditional link-faults may occur. In other words, there exists a path of odd length from 1 to 2n 1, excluding 3, between any two adjacent nodes in a faulty n-cube with 2n 5 conditional link-faults. In this paper, we are curi-ous whether paths of variable lengths still can be constructed to join two arbitrary nodes of distance greater than one. More precisely, we will show that a conditionally faulty n-cube, with 2n 5 faulty links, contains a fault-free path of length l be-tween any two nodes u and

v

of distance dP2 for each l satisfying d6_{l 6 2}n_{1 and 2jðl d}_Þ.

The rest of this paper is organized as follows. In Section2, basic deﬁnitions and notations are introduced. In Section3, the fault diameter of the n-cube is investigated. The partition of a conditionally faulty n-cube is presented in Section4. Fault-tolerant path embedding is shown in Section5. Finally, the conclusion is presented in Section6.

2. Preliminaries

Throughout this paper, we concentrate on loopless undirected graphs. For the graph deﬁnitions, we follow the ones given by Bondy and Murty[1]. A graph G consists of a node set VðGÞ and a link set EðGÞ that is a subset of fðu;

v

Þjðu;

v

Þ is an unor-dered pair of VðGÞg. Two nodes, u and

v

, of G are adjacent if ðu;

v

Þ 2 EðGÞ. Then u is a neighbor of

v

, and vice versa. A graph H is a subgraph of G if VðHÞ # VðGÞ and EðHÞ # EðGÞ. A graph G is bipartite if its node set can be partitioned into two disjoint partite sets, V0ðGÞ and V1ðGÞ, such that every link joins a node of V0ðGÞ and a node of V1ðGÞ.

A path P of length k from node x to node y in a graph G is a sequence of distinct nodes h

v

1;

v

2; . . . ;

v

kþ1i such that

v

1¼ x;

v

kþ1¼ y, and ð

v

i;

v

iþ1Þ 2 EðGÞ for every 1 6 i 6 k if k P 1. Moreover, a path of length zero consisting of a single node

x is denoted by hxi. For convenience, we write P as h

v

1; . . . ;

v

i;Q ;

v

j; . . . ;

v

kþ1i, where Q ¼ h

v

i; . . . ;

v

ji. The ith node of P is

de-noted by PðiÞ; i.e., PðiÞ ¼

v

i. We use ‘ðPÞ to denote the length of P. The distance between any two nodes, u and

v

, of G, denoted

by dGðu;

v

Þ, is the length of the shortest path joining u and

v

in G. The diameter of G, denoted by DðGÞ, is deﬁned to be

maxfdGðu;

v

Þ j u;

v

2 VðGÞg. A cycle is a path with at least three nodes such that the last node is adjacent to the ﬁrst one.

For clarity, a cycle of length k is represented by h

v

1;

v

2; . . . ;

v

k;

v

1i. A path (or cycle) in a graph G is a hamiltonian path (or

hamiltonian cycle) if it spans G. A bipartite graph is hamiltonian laceable[19]if there exists a hamiltonian path between any two nodes that are in different partite sets. Moreover, a hamiltonian laceable graph G is hyper-hamiltonian laceable

[15]if, for any node

v

2 ViðGÞ and i 2 f0; 1g, there exists a hamiltonian path of G f

v

g between two arbitrary nodes of

V1iðGÞ. Later Hsieh et al.[9]introduced strongly hamiltonian laceability. A hamiltonian laceable graph G is strongly

hamilto-nian laceable if there exists a path of length jVðGÞj 2 between any two nodes in the same partite set.

Let u ¼ bn1. . .bi. . .b0 be an n-bit binary string. For any j, 0 6 j 6 n 1, we use ðuÞj to denote the binary string

bn1. . . bj. . .b0. Moreover, we use ðuÞj to denote the bit bj of u. The Hamming weight of u, denoted by wHðuÞ, is

jf0 6 i 6 n 1 j ðuÞi¼ 1gj. The n-cube Qn consists of 2nnodes and n2n1 links. Each node corresponds to an n-bit binary

string. Two nodes, u and

v

, are adjacent if and only if

v

¼ ðuÞjfor some j and we call the link ðu; ðuÞjÞ j-dimensional. We deﬁne dimððu;

v

ÞÞ ¼ j if

v

¼ ðuÞj. The Hamming distance between u and

v

, denoted by hðu;

v

Þ, is deﬁned to be jf0 6 i 6 n 1 j ðuÞi–ð

v

Þigj. Hence two nodes, u and

v

, are adjacent if and only if hðu;

v

Þ ¼ 1. It is well known that Qnis a

bipartite graph with partite sets V0ðQnÞ ¼ fu 2 VðQnÞjwHðuÞ is even} and V1ðQnÞ ¼ fu 2 VðQnÞjwHðuÞ is odd}. Moreover, Qn

is both node-transitive and link-transitive[14]. Let Qj;i

n be a subgraph of Qninduced by fu 2 VðQnÞ j ðuÞj¼ ig for 0 6 j 6 n 1 and i 2 f0; 1g. Clearly, Q j;i

n is isomorphic to

Qn1. Then the node partition of Qninto subgraphs Qj;0n and Q j;1

n is called j-partition. The set of crossing links between Q j;0 n and Qj;1n, denoted by E j c¼ fðu;

v

Þ 2 EðQnÞ j u 2 VðQ j;0 nÞ;

v

2 VðQ j;1

nÞg, consists of all j-dimensional links of Qn. In order to clearly

indi-cate the faulty elements in graph G, we use FðGÞ to denote the set of all faulty elements in G. 3. Fault diameter of the n-cube

Let G be a graph. A faulty link (or faulty node) of G is a link (or node) that can be deleted from G. To be precise, the deletion of a subset Feof EðGÞ, denoted by G Fe, is the spanning subgraph of G obtained by deleting the links in Fefrom G; the

dele-tion of a proper subset Fvof VðGÞ, denoted by G Fv, is the subgraph containing the nodes of G not in Fvand the links of G not

incident with any node in Fv. By such deﬁnition, if a node is deleted from G, then all links incident with this node are deleted.

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say that

v

is a reachable neighbor of u if both

v

and ðu;

v

Þ are fault-free; otherwise,

v

is an unreachable neighbor of u. The following lemma is a basic property of Qn.

Lemma 1 [17]. For any two nodes, u and v, of Qn, there exist n internally node-disjoint paths joining u and v, hðu;

v

Þ of which are

of length hðu;

v

Þ and the other n hðu;

v

Þ of which are of length hðu;

v

Þ þ 2. The next corollary directly follows fromLemma 1.

Corollary 1. Let F be a set of n 1 node-faults and/or link-faults in Qn. For any pair u;

v

of distinct nodes in Qn F, then

dQnFðu;

v

Þ 6 hðu;

v

Þ þ 2.

Latiﬁ[12]investigated the fault diameter of Qnunder the assumption that every node has at least one fault-free neighbor.

The following theorem was proved in[12].

Theorem 1 [12]. Let F be a set of 2n 3 faulty nodes in Qnsuch that every node of Qnhas at least one fault-free neighbor. For any

pair u;

v

of distinct nodes in Qn F, then dQnFðu;

v

Þ 6 hðu;

v

Þ þ 4.

Although only node-faults are admitted by Latiﬁ[12], it is noticed that a similar result can be obtained when both node-faults and link-node-faults are involved. To be precise, we improveTheorem 1by proving the next corollary.

Corollary 2. Suppose that u and v are any two distinct nodes of Qn;n P 2. Let F be a set of utmost 2n 3 hybrid node-faults and/

or link-faults in Qnsuch that both u and v are fault-free with at least one reachable neighbor. Then

dQnFðu;

v

Þ

¼ n if jFj 6 2n 3; hðu;

v

Þ ¼ n; and n P 2; 6n þ 1 if jFj 6 2n 3; hðu;

v

_{Þ ¼ n 1; and n P 2;} 6_hðu;

v

_{Þ þ 4 if jFj 6 2n 3; hðu;}

v

_{Þ 6 n 2; and n P 3;} 6n if jFj ¼ 2n 4; hðu;

v

_{Þ ¼ n 2; and n – 4:} 8 > > > > < > > > > :

For clarity, we prove the the ﬁrst part ofCorollary 2in advance.

Proposition 1. Suppose that u and v are any two distinct nodes of Qnwith hðu;

v

Þ ¼ n. Let F be a set of 2n 3 hybrid node-faults

and/or link-faults in Qnsuch that both u and v are fault-free with at least one reachable neighbor. Then dQnFðu;

v

Þ ¼ n.

Proof. It is not difﬁcult to verify that this proposition holds for n ¼ 2. Hence, we only concern the case that n P 3. Let Iu¼ fi1; . . . ;ipg be a set of p distinct integers of f0; 1; . . . ; n 1g such that ðuÞi1; . . . ;ðuÞipare reachable neighbors of u. Similarly,

let Iv¼ fi01; . . . ;i 0

qg # f0; 1; . . . ; n 1g be a set of q distinct integers such that ð

v

Þ i0

1_{; . . . ;ð}

v

_Þi 0

q _{are reachable neighbors of}

v

_{. We}

distinguish the following two cases.

Case 1: Suppose that Iu\ Iv – ;. Let j 2 Iu\ Iv . Then we partition Qninto Qj;0n and Q j;1

n. For convenience, let F0¼ FðQj;0nÞ and

F1¼ FðQj;1n Þ. Since hðu;

v

Þ ¼ n, nodes u and

v

are located in different subcubes. Moreover, we have hðu; ð

v

Þ j

Þ ¼ n 1. By the pigeonhole principle, we have jF0j 6 n 2 or jF1j 6 n 2. Without loss of generality, we assume that jF0j 6 n 2. Moreover,

we assume that u 2 VðQj;0nÞ. ByLemma 1, Q j;0

n has at least one fault-free path L of length n 1 between u and ð

v

Þ j

. Hence, hu; L; ð

v

Þj;

v

i forms a fault-free path of length n between u and

v

.

Case 2: Suppose that Iu\ Iv ¼ ;. Since jFj ¼ 2n 3, we can conclude that 3 6 p þ q 6 n. Without loss of generality, we

assume that p P q. Thus, we have p P 2.

Suppose that n ¼ 3. We have p ¼ 2 and q ¼ 1. Let j 2 Iv. Without loss of generality, we assume that u 2 VðQj;0

nÞ. Obviously

Qj;0n is fault-free and it has a fault-free path L of length two between u and ð

v

Þ j

. Then hu; L; ð

v

Þj;

v

i is a fault-free path of length three.

Suppose that n P 4. Let j 2 Iu. Since Iu\ Iv ¼ ;; ðuÞjis a reachable neighbor of u whereas ð

v

Þjis an unreachable neighbor of

v

. Again, we assume that u 2 VðQj;0n Þ. Let F0¼ FðQj;0nÞ and F1¼ FðQj;1n Þ. If jF1j 6 n 2,Lemma 1ensures that Qj;1n has a

fault-free path R of length n 1 between ðuÞjand

v

. Hence, hu; ðuÞj;R;

v

i is a fault-free path of length n between u and

v

. Suppose that jF1j P n 1. Thus, we have jF0j þ jF \ Ejcj 6 n 2. Let eIv ¼ fk 2 Iv jðð

v

Þ

k

Þj2 NQnFðð

v

ÞkÞg, where NQnFðð

v

ÞkÞ

is the set of all reachable neighbors of ð

v

Þk.

Subcase 2.1: Suppose that eIv – ;. Let k 2 eIv andHbe a subgraph of Qninduced by fx 2 VðQnÞjðxÞj¼ ðuÞj;ðxÞk¼ ðuÞkg.

ThenHis an ðn 2Þ-cube inside Qj;0n. Because ð

v

Þ j

is an unreachable neighbor of

v

and it is outsideH, there are utmost n 3 faulty elements inH. ByLemma 1,Hhas a fault-free path L of length n 2 between u and ðð

v

ÞkÞj. So hu; L; ðð

v

ÞkÞj;ð

v

Þk;

v

i is a fault-free path of length n.

Subcase 2.2: Suppose that ~Iv ¼ ;. Let k12 Iv . Since jFj 6 2n 3 and p þ q 6 n, there exists an integer

k22 f0; 1; . . . ; n 1g fj; k1g such that ðð

v

Þk1Þk2is a reachable neighbor of ð

v

Þk1 and ððð

v

Þk1Þk2Þjis a reachable neighbor of

ðð

v

Þk1_Þk2_{. Let w ¼ ðð}

v

_Þk1_Þk2_and_X_{be a subgraph of Q}

ninduced by fx 2 VðQnÞ j ðxÞj¼ ðuÞj;ðxÞk1¼ ðuÞk1;ðxÞk2¼ ðuÞk2g. ThenXis

an ðn 3Þ-cube inside Qj;0n . Obviously, ðuÞ k1

, ð

v

Þj, and ðð

v

Þk1Þjare unreachable neighbors of u,

v

, and ð

v

Þk1, respectively. Since ðuÞk1;ð

v

Þj, and ðð

v

Þk1Þjare outsideX, there are utmost n 4 faulty elements inX. It follows fromLemma 1thatXhas a fault-free path L of length n 3 between u and ðwÞj. So hu; L; ðwÞj;w; ðwÞk2_{¼ ð}

v

_Þk1_;

v

_{i is a fault-free path of length n between u and}

v

_.

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Proof of Corollary 2. Now we concern that hðu;

v

Þ 6 n 1. The proof is by the induction on n. The result is true for n ¼ 2. As our inductive hypothesis, we assume that the result holds for Qn1with n P 3. Since hðu;

v

Þ 6 n 1, we partition Qnalong

some dimension j such that both u and

v

are in the same subcube. By transitivity, we assume that j ¼ 0 and u;

v

2 VðQ0;1n Þ. Let

Fi¼ FðQ0;inÞ for i 2 f0; 1g.

Case 1: Suppose that jF1j 6 2n 5 ¼ 2ðn 1Þ 3. First, we consider the case that both u and

v

have at least one reachable

neighbor in Q0;1n . Then it follows from the inductive hypothesis that dQnFðu;

v

Þ ¼ dQ0;1

n F1ðu;

v

Þ ¼ n 1 if

hðu;

v

Þ ¼ n 1; dQnFðu;

v

Þ 6 dQ0;1

nF1ðu;

v

Þ 6 n if hðu;

v

Þ ¼ n 2, and dQnFðu;

v

Þ 6 dQ0;1n F1ðu;

v

Þ 6 hðu;

v

Þ þ 4 if hðu;

v

Þ 6

n 3 for n P 4.

Now we consider the case that either u or

v

has no reachable neighbors in Q0;1n . Thus, we have jF1j P n 1 and

jF0j þ jF \ E0cj 6 n 2. Since n 1 6 jF1j 6 2n 5, we have n P 4. Without loss of generality, we assume that u has no

reachable neighbors in Q0;1

n . Accordingly, ðuÞ 0

is the unique reachable neighbor of u.

Suppose that hðu;

v

Þ ¼ n 1. Since hððuÞ0;

v

Þ ¼ n, it follows fromProposition 1that dQnFððuÞ0;

v

Þ ¼ n. Let P be a

fault-free path of length n between ðuÞ0and

v

. Obviously, we have u R VðPÞ. Hence hu; ðuÞ0;P;

v

i forms a fault-free path of length n þ 1.

Suppose that hðu;

v

Þ 6 n 2. If ð

v

Þ0 is a reachable neighbor of

v

, then it follows from Corollary 1 that d_Q0;0

n F0ððuÞ 0

;ð

v

Þ0Þ 6 hððuÞ0;ð

v

Þ0Þ þ 2 ¼ hðu;

v

Þ þ 2 since jF0j 6 n 2. Let H be a shortest path between ðuÞ0 and ð

v

Þ0 in

Q0;0n F0. Then hu; ðuÞ0;H; ð

v

Þ0;

v

i forms a fault-free path of length at most hðu;

v

Þ þ 4. When jFj ¼ 2n 4, we have

jF0j 6 n 3. Therefore, Q0;0n F0has a path H of length n 2 between ðuÞ0and ð

v

Þ0if hðu;

v

Þ ¼ n 2. Thus hu; ðuÞ0;H; ð

v

Þ0;

v

i

is a fault-free path of length n. On the other hand, if ð

v

Þ0 is an unreachable neighbor of

v

, then we have ð

v

Þ02 F or ð

v

;ð

v

Þ0Þ 2 F. ByLemma 1, Q0;0

n has n 1 internally node-disjoint paths L1; . . . ;Ln1between ðuÞ0and ð

v

Þ0. For clarity, Lican

be written as hðuÞ0;L0i;ðð

v

Þ 0

Þi;ð

v

Þ0i for 1 6 i 6 n 1. Let Ti¼ hðuÞ0;L0i;ðð

v

Þ 0

Þi;ð

v

Þi;

v

i with 1 6 i 6 n 1. Then fT1; . . . ;Tn1g is

a set of n 1 internally node-disjoint paths between ðuÞ0and

v

. We distinguish two subcases.

Subcase 1.1: One of fT1; . . . ;Tn1g, say Ti, is fault-free. Hence, hu; ðuÞ0;Ti;

v

i is a path of length at most hðu;

v

Þ þ 4 between

u and

v

. In particular, we consider the case that hðu;

v

Þ ¼ n 2. Clearly, n 2 paths of fT1; . . . ;Tn1g are of length n 1. When

n P 5, u and

v

have no common neighbors. Since ðfð

v

Þ0;ð

v

;ð

v

Þ0Þg [Sn1i¼1fðuÞ i

;ðu; ðuÞiÞgÞ \ ðSn1i¼1VðTiÞ [ EðTiÞÞ ¼ ;, at most

n 3 faults may appear on T1; . . . ;Tn1. Hence there exists a fault-free path Tkof fT1; . . . ;Tn1g such that ‘ðTkÞ ¼ n 1 if

n P 5. Then hu; ðuÞ0;Tk;

v

i is a fault-free path of length n.

Subcase 1.2: None of fT1; . . . ;Tn1g is fault-free. It is noticed that jFj ¼ 2n 3 in this subcase. Moreover, we claim that

hðu;

v

Þ ¼ 2. Because T1; . . . ;Tn1are internally node-disjoint and u has no reachable neighbors in Q0;1n , every of fT1; . . . ;Tn1g

contains exactly one faulty element. Since VðTiÞ \ VðQ0;1n Þ ¼ f

v

;ð

v

Þ i

g for 1 6 i 6 n 1, there exist two distinct integers t1and

t2;1 6 t1;t26n 1, such that FðT_t1_{Þ ¼ fð}

v

_Þt1_{g ¼ fðuÞ}t2_{g and FðT}_t2_{Þ ¼ fð}

v

_Þt2_{g ¼ fðuÞ}t1_{g. By transitivity, we assume that} t1¼ n 1 and t2¼ n 2. Again,Lemma 1ensures that Q0;1n has n 1 internally node-disjoint paths R1; . . . ;Rn1of length at

most four between u and

v

. For clarity, we can write Rias hu; R0i;ð

v

Þ i

;

v

i for 1 6 i 6 n 1. Thus, we have ‘ðRn2Þ ¼ ‘ðRn1Þ ¼ 2

and ‘ðRiÞ ¼ 4 for 1 6 i 6 n 3. Because ð

v

Þ0is an unreachable neighbor of

v

,

v

has a reachable neighbor in Q0;1n , say ð

v

Þ k

with some k 2 f1; . . . ; n 3g. To be precise, we write Rk¼ hu; xk;yk;ð

v

Þ

k

;

v

i and Lk¼ hðuÞ0;ðxkÞ0;ðykÞ 0

;ðð

v

ÞkÞ0;ð

v

Þ0i, where xk is

some neighbor of u and ykis a common neighbor of xkand ð

v

Þk.

Subcase 1.2.1: Suppose that ðð

v

ÞkÞ0 is an unreachable neighbor of ð

v

Þk. Let S_kð1Þ¼ hðuÞ0;ðxkÞ0, ðykÞ0i and

Sð2Þ_k ¼ hðykÞ 0

;yk;ð

v

Þ k

i. Because Tk has only one faulty element, S ð1Þ k is fault-free. Since ðVðS ð2Þ k Þ [ EðS ð2Þ k ÞÞ\

ðSi – kVðTiÞ [ EðTiÞÞ ¼ ;; Sð2Þk is also fault-free. Then hu; ðuÞ 0

;Sð1Þ_k ;ðykÞ 0

;Sð2Þ_k ;ð

v

Þk;

v

i is a fault-free path of length six.

Subcase 1.2.2: Suppose that ðð

v

ÞkÞ0 is a reachable neighbor of ð

v

Þk. Let H be the subgraph of Q0;0n induced by

fx 2 VðQ0;0n Þ j ðxÞp¼ ðuÞp;p 2 f1; . . . ; n 3g fkgg. Obviously,His isomorphic to Q3. Then we claim that jFðHÞj 6 2. Since

jF0j 6 n 2, this claim holds for n ¼ 4. In what follows, we concern that n P 5. It is easy to see that Lk;Ln2, and Ln1are

insideH. Moreover, we have ðVðTiÞ [ EðTiÞÞ \ ðVðHÞ [ EðHÞÞ ¼ fðuÞ0g for i 2 f1; . . . ; n 3g fkg. Since Ticontains one faulty

element for each 1 6 i 6 n 1, at least n 4 faulty elements are outsideH; i.e., jFðHÞj 6 2. Since hððuÞ0;ðð

v

ÞkÞ0Þ ¼ 3, it follows from Lemma 1 that H has a fault-free path S of length three between ðuÞ0 and ðð

v

ÞkÞ0. As a result, hu; ðuÞ0;S; ðð

_v

ÞkÞ0;ð

v

Þk;

v

i is a fault-free path of length six.

Case 2: Suppose that jF1j P 2n 4. Thus, we have jF0j þ jF \ E0cj 6 1.

Subcase 2.1: Suppose that ðuÞ0and ð

v

Þ0are reachable neighbors of u and

v

, respectively. Since jF0j 6 1, it follows from Lemma 1that Q0;0

n has a fault-free path L of length at most hðu;

v

Þ þ 2 between ðuÞ 0_{and ð}

_v

Þ0. Then hu; ðuÞ0

;L; ð

v

Þ0;

v

i is a fault-free path of length at most hðu;

v

Þ þ 4 between u and

v

. When jFj ¼ 2n 4, we have jF0j þ jF \ E0cj ¼ 0. Hence Q0;0n has a

path L of length hðu;

v

Þ between ðuÞ0and ð

v

Þ0. Then hu; ðuÞ0;L; ð

v

Þ0;

v

i is a fault-free path of length hðu;

v

Þ þ 2 between u and

v

.

Subcase 2.2: Suppose that ðuÞ0or ð

v

Þ0is an unreachable neighbor of u or

v

, respectively. It is noticed that jFj ¼ 2n 3 in this subcase. Since jF0j þ jF \ E0cj 6 1, we assume that ðuÞ

0

is an unreachable neighbor of u. If

v

is a reachable neighbor of u, then dQnFðu;

v

Þ ¼ 1. Otherwise, let ðuÞk be a reachable neighbor of u with some k 2 f1; . . . ; n 1g. Since

(5)

jF0j þ jF \ E0cj 6 1; ððuÞ k

Þ0is a reachable neighbor of ðuÞk. If ðuÞk–ð

v

Þk, then hððuÞ k

;

v

Þ ¼ hðu;

v

Þ 1. Obviously, ðuÞ0is not on any shortest path between ððuÞkÞ0and ð

v

Þ0. Thus, Q0;0n has a fault-free path L of length hðððuÞ

k

Þ0;ð

v

Þ0Þ ¼ hðu;

v

Þ 1 between ððuÞkÞ0 and ð

v

Þ0. Then hu; ðuÞk;ððuÞkÞ0;L; ð

v

Þ0;

v

Þ þ 2. If ðuÞk¼ ð

v

Þk, then hððuÞ

k

;

v

Þ ¼ hðu;

v

Þ þ 1. By Lemma 1, Q0;0n has a fault-free path L of length hðu;

v

Þ þ 1 between ððuÞ

k

Þ0 and ð

v

Þ0. Then hu; ðuÞk; ððuÞkÞ0;L; ð

v

Þ0;

v

Þ þ 4.

The proof is completed. h

The following theorem characterizes a property of shortest paths in a faulty n-cube.

Theorem 2. Let F be a set of 2n 5 faulty links in Qnsuch that every node of Qn F has at least two neighbors. Moreover, let j be

an integer of f0; 1; . . . ; n 1g such that both Qj;0 n and Q

j;1

n are conditionally faulty with 2n 7 or less faulty links. Suppose that u is

a node of Qj;0n and v is a node of Q j;1

n. Then there exists a shortest path P between u and v in Qn F such that Pcrosses the

dimension j exactly once. Proof. Since jFðQj;0

nÞj þ jFðQ j;1

nÞj 6 jFj ¼ 2n 5, we assume that jFðQ j;1

nÞj 6 n 3. Since ðuÞj–ð

v

Þj, every shortest path

between u and

v

crosses the dimension j an odd number of times. If there is a shortest path between u and

v

crossing the dimension j exactly once, the proof is done. Thus, we assume that one shortest path between u and

v

, namely P, crosses the dimension j more than once. Accordingly, the shortest path P can be represented as hu; P0;x1;ðx1Þj;

P1;ðx2Þj;x2;P2;x3;ðx3Þj; . . . ;xr;ðxrÞj;Pr;

v

i with odd integer r P 3. For convenience, let H ¼ hðx1Þj;P1;ðx2Þj; x2;P2;

x3;ðx3Þj; . . . ;xr;ðxrÞj;Pr;

v

i. By Corollary 1, we have dQj;1

nFðQj;1nÞððx1Þ

j

;

v

Þ 6 hððx1Þj;

v

Þ þ 2. Suppose that R is a shortest path

between ðx1Þjand

v

in Qj;1n FðQ j;1

nÞ. Then we have ‘ðHÞ 6 ‘ðRÞ. Since r P 3, we have ‘ðHÞ P hððx1Þj;

v

Þ þ 2 P ‘ðRÞ. As a result,

P_{¼ hu; P}

0;x1;ðx1Þj;R;

v

i happens to be a shortest path between u and

v

and it crosses the dimension j exactly once. h

The fault diameter of Qnis computed as follows.

Theorem 3. [12]Let F be a set of faulty nodes in Qnsuch that every node of Qnhas at least one fault-free neighbor. Then the

diameter of Qn F is computed as follows:

DðQn FÞ ¼ n if jFj 6 n 2; n þ 1 if jFj ¼ n 1; n þ 2 if jFj ¼ 2n 3: 8 > < > :

We improveTheorem 3by proving the next corollary.

Corollary 3. Let F be a set of hybrid node-faults and/or link-faults in Qn, n P 3, such that every node of Qn has at least one

reachable neighbor. Then DðQ4 FÞ ¼ 4 if jFj 6 2; DðQ4 FÞ ¼ 5 if jFj ¼ 3; DðQ4 FÞ ¼ 6 if jFj 2 f4; 5g. When n – 4,

DðQn FÞ ¼ n if jFj 6 n 2; n þ 1 if n 1 6 jFj 6 2n 4; n þ 2 if jFj ¼ 2n 3: 8 > < > : 0100 0101 1 1 1 0 0 1 1 0 0000 ₀₀₀₁ 0010 0011 1101 1110 1111 1000 1001 1010 1011 1100

Q

4

(6)

Proof. Suppose that n – 4. The result follows fromLemma 1,Corollary 2, andTheorem 3. Suppose that n ¼ 4. Applying

Lemma 1, Corollary 2, and Theorem 3, we also have DðQ4 FÞ ¼ 4 if jFj 6 2; DðQ4 FÞ ¼ 5 if jFj ¼ 3; DðQ4 FÞ 6 6 if

jFj ¼ 4, and DðQ4 FÞ ¼ 6 if jFj ¼ 5. Let F ¼ f0000; 0101; 0110; ð0111; 1111Þg. Then dQ4Fð0100; 0111Þ ¼ 6. SeeFig. 1.

There-fore, DðQ4 FÞ ¼ 6 if jFj ¼ 4. h

4. Partition of an n-cube with conditional link-faults

In this section, we propose a procedure to partition Qnwith 2n 5 conditional link-faults. Recall that a network is said to

be conditionally faulty if every node of this network is incident to at least two fault-free links. Suppose that Qn;n P 4, is

con-ditionally faulty with 2n 5 faulty links. For convenience, let F ¼ FðQnÞ and Fidenote the set of faulty i-dimensional links.

Since jFj ¼ 2n 5, there are utmost two nodes of Qnincident to n 2 faulty links. For any two distinct nodes, u and

v

, of Qn,

the procedure PartitionðQn;F; u;

v

Þ determines a dimension j according to the following rules:

(1) Suppose that there are exactly two nodes incident to n 2 faulty links. Then the two nodes must be connected by a faulty link ðw; ðwÞjÞ with some j 2 f0; 1; . . . ; n 1g. Obviously, both Qj;0n and Q

j;1

n are conditionally faulty with n 3

faulty links.

(2) Suppose that there is only one node, namely z, incident to n 2 faulty links. Let S ¼ f0 6 i 6 n 1 j ðz; ðzÞiÞ 2 Fg ¼ fk3; . . . ;kng and f0; 1; . . . ; n 1g S ¼ fk1;k2g. Then both Qi;0n and Q

i;1

n are conditionally faulty for each

i 2 S.

(2.1) If there exists a dimension j of S such that jFjj > 1, then we partition Qnalong dimension j. Otherwise, if there

exists a dimension j of S such that jFðQj;0nÞj jFðQ j;1

nÞj > 0, then we partition Qnalong dimension j. Obviously, both

Qj;0 n and Q

j;1

n contain 2n 7 or less faulty links.

(2.2) Suppose that jFij ¼ 1 and jFðQi;0nÞj jFðQ i;1

nÞj ¼ 0 for every i 2 S. That is, for any i 2 S, either jFðQ i;0

nÞj or jFðQ i;1 nÞj

remains 2n 6. Hence, for any ðx; yÞ 2 F fðz; ðzÞiÞ j i 2 Sg, we have ðxÞi¼ ðyÞi¼ ðzÞifor every i 2 S. That is, for

ðx; yÞ 2 F fðz; ðzÞiÞ j i 2 Sg, we have x; y 2 fz; ðzÞk1_;_ðzÞk2_;_ððzÞk1_Þk2_{g. Because both ðz; ðzÞ}k1_{Þ and ðz; ðzÞ}k2_{Þ are}

fault-free, it follows that F fðz; ðzÞiÞ j i 2 Sg # fððzÞk1_;_ððzÞk1_Þk2_{Þ; ððzÞ}k2_;_ððzÞk1_Þk2_{Þg. Since jF fðz; ðzÞ}i_{Þ j i 2 Sgj ¼ n}

3 6 2, we obtain n 2 f4; 5g. The faulty links are distributed as illustrated inFig. 2.

(2.2.1) If there exists a dimension j of S such that ðzÞjis neither u nor

v

, then we partition Qnalong dimension j.

(2.2.2) Otherwise, fu;

v

g equals to fðzÞij i 2 Sg; thus, we have n ¼ 4. In this case, we partition Q4along any

dimension j 2 S. Clearly, u and

v

belong to the same partite set of Q4.

(3) Suppose that every node is incident to utmost n 3 faulty links. Obviously, every ðn 1Þ-cube in Qnis conditionally

faulty. Let S ¼ f0 6 i 6 n 1 j Fi–;g.

(3.1) Suppose that jFjj P 2 with some j 2 S. Then both Qj;0n and Q j;1

(3.2) Suppose that jFij 6 1 for each i 2 S. Clearly we have 2n 5 ¼ jFj ¼ jSi2SFij ¼Pi2SjFij 6 n; i.e., n 6 5. Then a

dimension j of S can be chosen so that both Qj;0 n and Q

j;1

(3.2.1) When n ¼ 5, we claim that jFðQj;0_nÞj jFðQj;1_nÞj > 0 for some j 2 S. Let ei¼ ðbi4. . .bii. . .bi0;bi4. . . bii. . .bi0Þ

be an i-dimensional link of Q5for i 2 f0; 1; 2; 3; 4g. Suppose that F ¼ fe0;e1;e2;e3;e4g is a faulty set of

Q5 such that jFðQi;05Þj jFðQ i;1

5Þj ¼ 0 for each i 2 f0; 1; 2; 3; 4g. Then we have b0i¼ b1i¼ b2i¼ b3i¼ b4i

for each i 2 f0; 1; 2; 3; 4g; i.e., all faulty links are incident with an identical node. This contradicts the assumption that every node is incident to utmost n 3 faulty links.

(3.2.2) Similarly, there exists an integer j 2 S such that jFðQj;0 4Þj jFðQ

j;1 4Þj > 0.

In summary, the proposed procedure determines a j-partition of Qnsuch that both Qj;0n and Q j;1

n are conditionally faulty

with jFðQj;0 nÞj þ jFðQ j;1 nÞj 6 2n 6. Q₅ z k ( )z 3 k ( )z 4 k ( )z 5 k ( )z 1 k ( )z 2 k ( )z 1 ( )k2 Q₄ z k ( )z 3 k ( )z 4 k ( )z 1 k ( )z 2 k ( )z 1 ( )k2

(a)

(b)

(7)

5. Path embedding in hypercubes

The following theorems were proved by Tsai[20]and Xu[24].

Theorem 4 [20]. Let n P 3. Suppose that F # EðQnÞ is a set of utmost n 2 faulty links. Then Qn F is hamiltonian laceable and

strongly hamiltonian laceable.

Theorem 5 [20]. Let n P 3. Suppose that F # EðQnÞ is a set of utmost n 3 faulty links. Then Qn F is hyper-hamiltonian

laceable.

Theorem 6 [24]. Let F be a set of n 2 faulty links in Qn(n P 2). Suppose that u and v are any two different nodes of Qn F.

Then Qn F contains a path of length l between u and v for every l satisfying dQnFðu;

v

Þ 6 l 6 2

n

1 and 2jðl dQnFðu;

v

ÞÞ.

As Tsai[21] showed, an n-cube with 2n 5 conditional link-faults is hamiltonian laceable and strongly hamiltonian laceable.

Theorem 7 [21]. Let F be a set of faulty links in Qn(n P 3) such that every node of Qn F has at least two neighbors. Then Qn F

is hamiltonian laceable and strongly hamiltonian laceable if jFj 6 2n 5. To prove our main result, we need the next two lemmas.

Lemma 2 [21]. Assume that n P 2. Let x and u be two distinct nodes of V0ðQnÞ; let y and v be two distinct nodes of V1ðQnÞ. Then

there exist two node-disjoint paths P1and P2such that the following conditions are satisﬁed: (1) P1joins x to y, (2) P2joins u to v,

and (3) VðP1Þ [ VðP2Þ ¼ VðQnÞ.

Lemma 3. Let v be any node of Qnðn P 3Þ and let ðw; bÞ be any link of Qn f

v

g. For every odd integer l in the range from 1 to

2n

3; Qn f

v

g has a path of length l between w and b.

Proof. Since Qnis node-transitive, we assume that

v

¼ 0n. We prove this lemma by the induction on n. The induction base

depends on Q3. With the link-transitivity, the required paths are listed inTable 1.

When n P 4, we assume that the result is true for Qn1. Then we partition Qnalong dimension p other than dimððw; bÞÞ.

Obviously,

v

is located in Qp;0n .

Case 1: Suppose that ðw; bÞ is in Qp;0n . By the inductive hypothesis, Q p;0

n f

v

g has a path of odd length l0between w and b

for any odd integer l0from 1 to 2n1 3. Let H be a path of length 2n1 3 between w and b in Qp;0n f

v

g. Since 2n1 3 > 1,

we can represent H as hw; u; H0;bi. ByTheorem 6, Qp;1n has a path H1of odd length l1 between ðwÞpand ðuÞpfor any odd

integer l1from 1 to 2n1 1. As a result, hw; ðwÞp;H1;ðuÞp;u; H0;bi is a path of odd length 2n1 2 þ l1, in the range from

2n1 1 to 2n 3.

Case 2: Suppose that ðw; bÞ is in Qp;1n . ByTheorem 6, Q p;1

n has a path of odd length l1between w and b for any odd integer l1

from 1 to 2n1_{1. Let H be a path of length 2}n1_{1 between w and b in Q}p;1

n . Then we can choose a link ðx; yÞ on H such that

v

RfðxÞp;ðyÞpg. Hence, we can represent H as hw; H01;x; y; H001;bi. By the inductive hypothesis, Q p;0

n f

v

g has a path H0of odd

length l0between ðxÞpand ðyÞpfor any odd integer l0from 1 to 2n1 3. As a result, hw; H01;x; ðxÞ p

;H0;ðyÞp;y; H001;bi is a path of

odd length 2n1þ l0, in the range from 2n1þ 1 to 2n 3. h

As Shih et al.[18]showed, any fault-free link of Qnlies on a cycle of even length from 6 to 2nwhen up to 2n 5

condi-tional link-faults may occur.

Theorem 8 [18]. Let F be a set of 2n 5 faulty links in Qnsuch that every node of Qn F has at least two neighbors. Suppose that

u and v are any two adjacent nodes of Qn F. Then Qn F contains a path of odd length l between u and v if l is in the range from 1

to 2n 1 excluding 3.

In the following discussion, we focus on constructing paths between any two nodes with distance greater than one. Theorem 9. Let F be a set of 2n 5 faulty links in Qnðn P 3Þ such that every node of Qn F has at least two neighbors. Suppose

that u and v are two arbitrary nodes of Qn F with distance d¼ dQnFðu;

v

Þ P 2. Then Qn F contains a path of length l between

Table 1

The paths of variable lengths between w and b in Q3 f000g.

ðw; bÞ ¼ ð011; 001Þ h011; 111; 101; 001i; h011; 111; 110; 100; 101; 001i ðw; bÞ ¼ ð011; 111Þ h011; 001; 101; 111i; h011; 001; 101; 100; 110; 111i ðw; bÞ ¼ ð101; 001Þ h101; 111; 011; 001i; h101; 100; 110; 111; 011; 001i ðw; bÞ ¼ ð101; 100Þ h101; 111; 110; 100i; h101; 111; 011; 010; 110; 100i ðw; bÞ ¼ ð101; 111Þ h101; 100; 110; 111i; h101; 100; 110; 010; 011; 111i

(8)

u and v for every integer l satisfying both d₆_{l 6 2}n

1 and 2jðl dÞ, where expression 2jðl dÞ means that l d 0 ðmod 2Þ.

Proof. Applying procedure Partition(Qn, F, u,

v

), we can determine a j-partition of Qnsuch that both Qj;0n and Q j;1

n are

condi-tionally faulty with jFðQj;0 nÞj þ jFðQ

j;1

nÞj 6 2n 6. As a result, the proof can proceed by the induction on n. The induction base,

depending upon Q3, follows fromTheorem 6. As our inductive hypothesis, we assume that the result holds for Qn1when

n P 4.

Case I: Suppose that u and

v

are in the different partite sets of Qn. Without loss of generality, we assume that u 2 V0ðQnÞ

and

v

2 V1ðQnÞ. ByTheorem 7, Qn F is hamiltonian laceable. Moreover, a shortest path between u and

v

can be easily

obtained by a simple breadth-ﬁrst search. Therefore, we mainly concentrate on the paths of odd lengths in the range from dþ 2 to 2n 3.

Subcase I.1: Suppose that jFðQj;0n Þj 6 2n 7 and jFðQ j;1

nÞj 6 2n 7. Without loss of generality, we assume that

jFðQj;0n Þj P jFðQ j;1

n Þj; thus, jFðQ j;1

n Þj 6 n 3.

Subcase I.1.1: Suppose that both u and

v

are in Qj;0

n. By the inductive hypothesis, Q j;0 n FðQ

j;0

n Þ contains a path H0of length

2n1 1 between u and

v

. Let A ¼ fðH0ðiÞ; H0ði þ 1ÞÞ j 1 6 i 6 2n1;i 1 ðmod 2Þg be a set of disjoint links on H0. Since

jAj ¼ d2n1₁

2 e > 2n 5 for any n P 4, there exists a link ðw; bÞ of A such that ðw; ðwÞ j

Þ; ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. Hence, H0can be written as hu; H00;w; b; H000;

v

i. Since jFðQ

j;1

n Þj 6 n 3, it follows fromTheorem 6that Q j;1 n FðQ

j;1

n Þ contains a

path H1 of odd length l1 between ðwÞj and ðbÞj for any odd integer l1 from 1 to 2n1 1. As a result,

hu; H00;w; ðwÞ j

;H1;ðbÞj;b; H000;

v

i is a path of odd length 2n1þ l1, in the range from 2n1þ 1 to 2n 1. See Fig. 3a for

illustration.

The paths of lengths less than 2n1þ 1 can be obtained as follows. ByCorollary 2, we have d¼ dQnFðu;

v

Þ 6 hðu;

v

Þ þ 4

and d_Qj;0

nFðQj;0nÞðu;

v

Þ 6 hðu;

v

Þ þ 4. By the inductive hypothesis, Q j;0 n FðQ

j;0

n Þ has a path T0of length l0between u and

v

for any

odd integer l0 in the range from dQj;0

nFðQj;0nÞðu;

v

Þ to 2

n1_{1. If d}

¼ hðu;

v

Þ or d¼ hðu;

v

Þ þ 4, then d_Qj;0

nFðQj;0nÞðu;

v

Þ ¼ d

. Otherwise, if d¼ hðu;

v

nFðQj;0nÞðu;

v

Þ 6 d

þ 2.

v

are in Qj;1n . Since jFðQ j;1

nÞj 6 n 3, it follows from Corollary 1 that

d6_d

Qj;1

nFðQj;1nÞðu;

v

Þ 6 hðu;

v

Þ þ 2. Thus, there exists a shortest path between u and

v

in Qn F such that it does not cross

the dimension j. By inductive hypothesis, Qj;1n FðQ j;1

nÞ contains a path T1of odd length l1between u and

v

for each odd integer

l1 from d to 2n1 1. Let T1 be a path of length 2n1 1 between u and

v

in Qj;1n FðQj;1nÞ. Moreover, let

A ¼ fðT1ðiÞ; T1ði þ 1ÞÞ j 1 6 i 6 2n1;i 1 ðmod 2Þg be a set of disjoint links on T1. Since jAj ¼ d2 n1

1

2 e > 2n 5 for n P 4,

there exists a link ðw; bÞ of A such that ðw; ðwÞjÞ; ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. Hence, T1 can be written as

hu; T01;w; b; T 00

1;

v

i. Since jFðQ j;0

n Þj 6 2n 7, it follows fromTheorem 8that Q j;0 n FðQ

j;0

nÞ contains a path T0of odd length l0

between ðwÞj and ðbÞj for any odd integer l0 in the range from 1 to 2n1 1 excluding 3. As a result,

hu; T01;w; ðwÞ j

;T0;ðbÞj;b; T001;

v

i is a path of odd length 2n1þ l0, in the range from 2n1þ 1 to 2n 1 excluding 2n1þ 3. See Fig. 3b for illustration.

Q

_nj,0 j (w) j (b)

T

0 w b u v

Q

_nj,1

T

1

T

1

(b)

Q

₄j,0 j (w) j (b) w b u v

Q

₄j,1

T

1

T

1

(c)

j (w) ( )k j (b) ( )k

Q

₅j,0 j (w) j (b) w_b u v

Q

₅j,1

R

1

R

1

(d)

x y

_R

1 j (y) j (x) w b u v

Q

_nj,0

Q

_nj,1

H

0

H

0 _(w)j j (b)

_H

1

(a)

x u v

Q

_nj,0

Q

_nj,1

P

0

(e)

j (x)

P

1 x u v

Q

_nj,0

Q

_nj,1

P

0

(f)

j (x)

P

1

(9)

The path of length 2n1_{þ 3 is discussed as follows. When n ¼ 4, we have jFðQ}j;0

nÞj 6 1. Thus, there exists an integer k of

f0; 1; 2; 3g fj; dimððw; bÞÞg such that ððwÞj;ððwÞjÞkÞ, ððbÞj;ððbÞjÞkÞ, and ðððwÞjÞk;ððbÞjÞkÞ are all fault-free. Hence, hu; T01;w; ðwÞ

j

;ððwÞjÞk;ððbÞjÞk, ðbÞj;b; T00

1;

v

i is a path of length 11. See Fig. 3c for illustration. When n P 5, we have

jAj jFj ¼ jAj ð2n 5Þ ¼ d2n11

2 e ð2n 5Þ P 2. Thus, there is a link ðx; yÞ of A, other than ðw; bÞ, such that ðx; yÞ and ðw; bÞ

have no shared endpoints and ðx; ðxÞjÞ, ðy; ðyÞjÞ, and ððxÞj;ðyÞjÞ are all fault-free. Without loss of generality, T1can be written as

hu; R0 1;w; b; R001;x; y; R0001;

v

i. Hence, hu; R01;w; ðwÞ j ;ðbÞj;b; R00 1;x; ðxÞ j , ðyÞj;y; R000

1;

v

i is a path of length 2n1þ 3. SeeFig. 3d.

Subcase I.1.3: Suppose that u is in Qj;0n and

v

is in Q j;1

n. ByTheorem 2, we have a shortest path Pbetween u and

v

in Qn F

such that Pcrosses the dimension j exactly once. Thus, Pcan be represented as hu; P0;x; ðxÞj;P1;

v

i, where P0is a shortest

path joining u to some node x in Qj;0n FðQ j;0

nÞ and P1is a shortest path joining ðxÞjto

v

in Qj;1n FðQ j;1

n Þ. SeeFig. 3e and f for

illustration.

Subcase I.1.3.1: Suppose that ‘ðP0Þ > 0 and ‘ðP1Þ > 0. ByTheorem 6, Qnj;1 FðQj;1nÞ contains a path T1of length l1between

ðxÞjand

v

for each l1satisfying ‘ðP1Þ 6 l162n1 1 and 2jðl1 ‘ðP1ÞÞ. Suppose that ‘ðP0Þ ¼ 1. It follows fromTheorem 8that

Qj;0n FðQ j;0

n Þ contains a path T0of odd length l0 between u and x for any odd integer l0in the range from 1 to 2n1 1

excluding 3. Suppose that ‘ðP0Þ > 1. By the inductive hypothesis, Qj;0n FðQ j;0

nÞ contains a path T0of length l0between u and x

for each l0satisfying ‘ðP0Þ 6 l062n1 1 and 2jðl0 ‘ðP0ÞÞ. As a result, hu; T0;x; ðxÞj;T1;

v

i is a path of odd length l0þ l1þ 1,

in the range from dto 2n 3.

Subcase I.1.3.2: Suppose that ‘ðP0Þ ¼ 0 or ‘ðP1Þ ¼ 0. Since d¼ dQnFðu;

v

Þ > 1, we have u – x or

v

–ðxÞj. With symmetry,

we assume that ‘ðP0Þ ¼ 0. By the inductive hypothesis, Qj;1n FðQ j;1

nÞ contains a path T1of even length l1between ðxÞjand

v

for each even integer l1from ‘ðP1Þ to 2n1 2. As a result, hu ¼ x; ðxÞj;T1;

v

i is a path of odd length l1þ 1 in the range from

‘ðP1Þ þ 1 ¼ dto 2n1 1.

The paths of odd lengths in the range from 2n1þ 1 to 2n 1 are constructed as follows. Since jV1ðQj;0nÞj ¼ 2n2>2n 5 for

n P 4, we can choose a node y from V1ðQj;0nÞ such that ðy; ðyÞ j

Þ is fault-free. Let R0be a path joining u to y in Qj;0n FðQ j;0 n Þ and R1

be a path joining ðyÞjto

v

in Qj;1n FðQ j;1

n Þ. Similar to Subcase I.1.3.1, H ¼ hu; R0;y; ðyÞj;R1;

v

i is a path of any odd length in the

range from d0¼ d_Qj;0 nFðQ j;0 nÞðu; yÞ þ dQ j;1 nFðQ j;1 nÞððyÞ j ;

v

Þ þ 1 to 2n 1. ByCorollary 3, we have d06_{ðn þ 1Þ þ ðn 1Þ þ 1 6 2}n1_{þ 1} for n P 4. That is, H can be a path of any odd length in the range from 2n1þ 1 to 2n 1.

Subcase I.2: Suppose that jFðQj;0nÞj ¼ 2n 6 or jFðQ j;1

n Þj ¼ 2n 6. Without loss of generality, we assume that

jFðQj;0nÞj ¼ 2n 6. Thus, Q j;1

n is fault-free. By procedure PartitionðQn;F; u;

v

Þ, the faulty links are distributed as shown inFig. 2.

v

are in Qj;0n . Let ðw; bÞ be a faulty link of Q j;0

n such that both ðw; ðwÞ j

Þ and ðb; ðbÞjÞ are fault-free. For convenience, let F0¼ FðQj;0nÞ fðw; bÞg. By the inductive hypothesis, Q

j;0

n F0has a path Plof odd length l

between u and

v

for any odd integer l in the range from d_Qj;0

nF0ðu;

v

Þ to 2

n1_{1. If ðw; bÞ is on P}

l, we write Plas hu; P0l;w; b; P 00 l;

v

i

and deﬁne ePl¼ hu; P0l;w; ðwÞ j

;ðbÞj;b; P00

l;

v

i. Otherwise, Plcan be written as hu; P0l;x; y; P00l;

v

i, where ðx; yÞ is a link on Plsuch that

both ðx; ðxÞjÞ and ðy; ðyÞjÞ are fault-free. Similarly, we deﬁne ePl¼ hu; P0l;x; ðxÞ j

;ðyÞj;y; P00l;

v

i. Then ePlis a path of length l þ 2. By Corollary 2, we have d

¼ dQnFðu;

v

Þ 6 hðu;

v

Þ þ 4 and dQj;0

nF0ðu;

v

Þ 6 hðu;

v

Þ þ 4. First, if d

¼ hðu;

v

Þ or d¼ hðu;

v

Þ þ 4, then we have d¼ d_Qj;0

nF0ðu;

v

Þ and thus l ranges from d

to 2n1 1. Next, if d¼ hðu;

v

Þ þ 2 ¼ d_Qj;0

nF0ðu;

v

Þ, then l ranges from d

to 2n1 1. Finally, if d¼ hðu;

v

Þ þ 2 and d_Qj;0

nF0ðu;

v

Þ ¼ hðu;

v

Þ þ 4, then l ranges from d

þ 2 to 2n1 1. For the ﬁnal case, a shortest path between u and

v

in Qn F can be constructed by a breadth-ﬁrst search. In summary, the paths of odd lengths

from dþ 2 to 2n1þ 1 are constructed.

By Theorem 6, Qj;1n contains a path T1 of length l1 between ðwÞj and ðbÞj for each odd integer l1 from 1 to 2n1 1.

Similarly, Qj;1n contains a path R1 of length l1 between ðxÞj and ðyÞj for each odd integer l1 from 1 to 2n1 1. Thus,

hu; P0₂n1₁;w; ðwÞ j

;T1;ðbÞj;b; P00₂n1₁,

v

i (or hu; P0₂n1₁;x; ðxÞ j

;R1;ðyÞj;y; P₂00n1₁;

v

i) is a path of length 2n1þ l1, in the range

from 2n1þ 1 to 2n 1.

v

are in Qj;1n . Let ðw; ðwÞ i

Þ be a faulty link in Qj;0n such that both ðw; ðwÞ j

Þ and ððwÞi;ððwÞiÞjÞ are fault-free. Since d¼ dQnFðu;

v

Þ > 1, we assume that ðwÞjis different from u and

v

. Moreover, since n P 4, we

assume that t 2 f0; 1; . . . ; n 1g fj; ig. Let X ¼ fððwÞj;ððwÞjÞkÞ j k R fi; j; tgg. Since jXj ¼ n 3, our inductive hypothesis ensures that Qj;1n X contains a path T1of odd length l1between u and

v

for any odd integer l1satisfying d6l162n1 1. Let

T1denote a path of length 2n1 1 between u and

v

in Qj;1n X. It is noted that ððwÞ j ;ððwÞjÞiÞ is on T1. Hence, T1 can be represented as hu; T0 1;ðwÞ j ;ððwÞjÞi;T00 1;

v

i. ByTheorem 8, Q j;0 n ðFðQ j;0 n Þ fðw; ðwÞ i

ÞgÞ contains a path T0 of odd length l0

between w and ðwÞifor 5 6 l06₂n1_{1. As a result, hu; T}0

1;ðwÞ j

;w; T0;ðwÞi;ððwÞjÞi;T001;

v

i is a path of odd length 2n1þ l0, in

the range from 2n1_{þ 5 to 2}n

1. SeeFig. 4a for illustration.

Let T0 denote the longest path between w and ðwÞi in Qj;0n ðFðQ j;0

n Þ fðw; ðwÞ i

ÞgÞ. Moreover, let A ¼ fðT0ðkÞ; T0ðk þ 1ÞÞ j 1 6 k 6 2n1;k 1 ðmod 2Þg be a set of disjoint links on T0. The paths of lengths 2n1þ 1 and

(10)

(a) Since jAj ¼ d2n1₁

2 e > 3 for n P 4, there exists a link ðx; yÞ of A such that both F \ fðx; ðxÞ j

Þ; ðy; ðyÞjÞg ¼ ; and fðxÞj;ðyÞjg \ fu;

v

g ¼ ; are satisﬁed. Without loss of generality, we assume that x 2 V0ðQnÞ. ByLemma 2, there exist

two node-disjoint paths P1 and P2 in Qj;1n such that (i) P1 joins u to ðxÞj, (ii) P2 joins ðyÞj to

v

, and (iii)

VðP1Þ [ VðP2Þ ¼ VðQj;1nÞ. As a result, hu; P1;ðxÞj;x; y; ðyÞj;P2;

v

i is a path of length 2n1þ 1. SeeFig. 4b for illustration.

(b) We write T0 as hw ¼ x0;x1; . . . ;x2n1₁¼ ðwÞii. Then we can choose a pair of nodes from ffx0;x3g; fx1;x4g; fx2;x5gg,

namely fxk;xkþ3g, such that both F \ fðxk;ðxkÞjÞ; ðxkþ3;ðxkþ3ÞjÞg ¼ ; and jfðxkÞj;ðxkþ3Þjg \ fu;

v

gj 6 1 are satisﬁed.

(b.1) Suppose that xk2 V0ðQnÞ. If jfðxkÞj;ðxkþ3Þjg \ fu;

v

gj ¼ 0,Lemma 2ensures that Qj;1n has two node-disjoint paths

P1 and P2 such that (i) P1 joins u to ðxkÞj, (ii) P2 joins ðxkþ3Þj to

v

, and (iii) VðP1Þ [ VðP2Þ ¼ VðQj;1nÞ. Hence,

hu; P1;ðxkÞj;xk;xkþ1;xkþ2;xkþ3;ðxkþ3Þj;P2;

v

i is a path of length 2n1þ 3. If jfðxkÞj;ðxkþ3Þjg \ fu;

v

gj ¼ 1, we assume

that ðxkÞj¼

v

. ByTheorem 5, Qj;1n f

v

g has a hamiltonian path H1joining u to ðxkþ3Þj. Then hu; H1;ðxkþ3Þj;xkþ3,

xkþ2;xkþ1;xk;ðxkÞj¼

v

i is a path of length 2n1þ 3. SeeFig. 4c.

(b.2) Suppose that xk2 V1ðQnÞ. The required paths can be obtained similarly.

Subcase I.2.3: Suppose that u is in Qj;0n and

v

is in Q j;1

n . If ðu; ðuÞ j

Þ is fault-free, the shortest path between u and

v

can be of the form hu; ðuÞj;P1;

v

i, where P1is a shortest path joining ðuÞjto

v

in Qj;1n . By the inductive hypothesis, Q

j;1

n contains a path T1

of even length l1between ðuÞjand

v

for any even integer l1from d_Qj;1 nððuÞ

j

;

v

Þ ¼ d 1 to 2n1 2. Then hu; ðuÞj;T1;

v

i is a path

of odd length l1þ 1 in the range from dto 2n1 1. On the other hand, if ðu; ðuÞjÞ is faulty, we choose a neighbor of u, namely

x, in Qj;0n FðQ j;0

nÞ. Obviously, we have either hððxÞ j

;

v

Þ ¼ hðu;

v

Þ 2 or hððxÞj;

v

Þ ¼ hðu;

v

Þ. Let R1be a shortest path joining ðxÞj

to

v

in Qj;1n . Then hu; x; ðxÞ j

;R1;

v

i is a path of length hðu;

v

Þ or hðu;

v

Þ þ 2. Thus, we have d6hðu;

v

Þ þ 2. ByTheorem 6, Qj;1n

has a path T1of length l1between ðxÞjand

v

for any odd integer l1from hððxÞj;

v

Þ to 2n1 1. Then hu; x; ðxÞj;T1;

v

i is a path of

odd length l1þ 2 in the range from dþ 2 to 2n1þ 1.

The paths of lengths greater than 2n1_{1 can be obtained as follows. Since jFðQ}j;0

nÞj ¼ 2n 6, the j-partition determined

by Partition ðQn;F; u;

v

Þ guarantees that link ð

v

;ð

v

Þ j

Þ is fault-free if hðu;

v

Þ is odd. (See (2.2) in Section4). Let ðw; bÞ be a faulty link in Qj;0n such that both ðw; ðwÞ

j

Þ and ðb; ðbÞjÞ are fault-free. By the inductive hypothesis, Qj;0n ðFðQ j;0

nÞ fðw; bÞgÞ contains

a path H0of length 2n1 2 between u to ð

v

Þj. Three subcases are distinguished.

Subcase I.2.3.1: Suppose that ðw; bÞ is not located on H0. SeeFig. 4d. We choose a link ðx; yÞ on H0such that ðx; ðxÞjÞ and

ðy; ðyÞjÞ are fault-free and ððxÞj;ðyÞjÞ is not incident with

v

. Thus, H0can be represented as hu; H00;x; y; H000;ð

v

Þ j

i. ByLemma 3, Qj;1n f

v

g contains a path T1of odd length l1between ðxÞjand ðyÞjfor any odd integer l1from 1 to 2n1 3. Consequently,

hu; H00;x; ðxÞ j

;T1;ðyÞj;y; H000;ð

v

Þ j

;

v

i is a path of odd length 2n1þ l1, in the range from 2n1þ 1 to 2n 3.

Subcase I.2.3.2: Suppose that ðw; bÞ is located on H0 and ðw; bÞ is not incident with ð

v

Þj. SeeFig. 4e. Thus, H0can be

represented as hu; H0

0;w; b; H000;ð

v

Þ j

i. ByLemma 3, Qj;1n f

v

g contains a path T1 of odd length l1between ðwÞjand ðbÞjfor

1 6 l162n1_{3. Hence, hu; H}0

0;w; ðwÞ j

;T1;ðbÞj;b; H000;ð

v

Þ j

;

v

i is a path of odd length 2n1þ l1, in the range 2n1þ 1 to 2n 3.

Subcase I.2.3.3: Suppose that ðw; bÞ is located on H0and ðw; bÞ is incident with ð

v

Þj. SeeFig. 4f. Let w ¼ ð

v

Þj. Thus, H0can

be represented as hu; H0

0;b; w ¼ ð

v

Þ j

i. ByTheorem 6, Qj;1n contains a path T1of odd length l1between ðbÞjand

v

for any odd

integer l1 satisfying 1 6 l162n1 1. Then hu; H00;b; ðbÞ j

;T1;

v

i is a path of odd length 2n1þ l1 2, in the range from

2n1_{1 to 2}n 3.

H

0

H

0

H

0

Q

_n

j,0

j

(w)

T

0

w

u

v

Q

_n

j,1

T

1

T

1

(a)

j

(w)

(

)

i i

(w)

-X

w

u

v

Q

_n

j,0

Q

_n

j,1

(e)

j

(v)

b

j

(w)

j

(b)

T

1

u

v

Q

_n

j,0

Q

_n

j,1

(f)

j

(v)

b

T

1 j

(b)

Q

_n

j,0

j

(x)

T

0

w

u

v

Q

_n

j,1

(b)

i

(w)

j

(y)

P

2

P

1

x

y

H

0

x

u

v

Q

_n

j,0

Q

_n

j,1

(d)

j

(v)

y

j

(x)

j

(y)

H

0

T

1

Q

_n

j,0

w

u

v

Q

_n

j,1

(c)

i

(w)

H

1

x

k

x

k+1

T

0

x

k+2

x

k+3 j

x

k+3

(

)

(11)

Case II: Suppose that u and

v

belong to the same partite set of Qn. This case is similar to Case I and the details are

described inAppendix A. h

6. Conclusion

Fault tolerance is an important research issue in the area of interconnection networks. Since linear array and rings are two of the most fundamental structures, the node-fault and link-fault tolerance are widely investigated for path embedding in var-ious kinds of network topologies. By induction, we show that a conditionally faulty Qn, with 2n 5 faulty links, has a fault-free

path of odd (resp. even) length in the range from d_{to 2}n

1 between two arbitrary nodes of odd (resp. even) distance d. Let PrðnÞ denote the probability that every node of an n-cube containing 2n 5 faulty links is incident to at least two fault-free links. Then PrðnÞ is computed as follows: PrðnÞ ¼ 1 if n ¼ 3; PrðnÞ ¼ 1 2nð2n5nÞ

n2n1 2n5 if n ¼ 4; PrðnÞ ¼ 1 2 n n2n1 n n5 þ2n n n1 ð Þ n2n1 n n4 n2n1 2n5

if n P 5. One can verify that PrðnÞ approaches to 1 as n increases. Thus, the assumption of con-ditional link-faults is probabilistically reasonable.

Let u be any node of Qnand let

v

¼ ððuÞ0Þ1. Suppose that F ¼ fðu; ðuÞiÞ j 2 6 i 6 n 1g [ fð

v

;ð

v

ÞiÞ j 2 6 i 6 n 1g is a set of

2n 4 faulty links in Qn. Obviously, Qn F has no hamiltonian paths joining u and ðuÞ 1

. That is, an n-cube with 2n 4 or more conditional link-faults is likely to have no paths of some speciﬁc lengths. In this sense, our result is optimal. A number of researchers[5,8,10,22,23]addressed the fault-tolerant hamiltonicity (or hamiltonian connectivity) in some special classes of network topologies under the consideration of conditional fault model. For example, the crossed cube[3], which is a var-iation of hypercubes, possesses some properties superior to the hypercube. Fu [6]showed that a conditionally faulty n-dimensional crossed cube contains a fault-free hamiltonian cycle even if it has 2n 5 faulty links. Hence, it is intriguing to study fault-tolerant path embedding on crossed cubes under the assumption of conditional faults.

Acknowledgement

The authors would like to express the immense gratitude to the anonymous referees for their insightful comments that make this paper more precise.

Appendix A. Case II in proof ofTheorem 9

Case II: Suppose that u and

v

belong to the same partite set of Qn. Thus, the distance d

between u and

v

is even. Without loss of generality, we assume that u;

v

2 V0ðQnÞ. ByTheorem 7, Qn F is strongly hamiltonian laceable. Moreover, a shortest

path between u and

v

can be obtained by a breadth-ﬁrst search. Hence, we concentrate on the paths of even lengths in the range from dþ 2 to 2n 4.

Subcase II.1: Suppose that jFðQj;0nÞj 6 2n 7 and jFðQ j;1

jFðQj;0nÞj P jFðQ j;1

nÞj. Thus, jFðQ j;1

nÞj 6 n 3.

Subcase II.1.1: Suppose that both u and

v

are in Qj;0

n. By the inductive hypothesis, Q j;0 n FðQ

j;0

nÞ has a path H0of length

2n1

2 between u and

v

. Let A ¼ fðH0ðiÞ; H0ði þ 1ÞÞ j 1 6 i 6 2n1 1; i 1 ðmod 2Þg be a set of disjoint links on H0. First,

sup-pose that jFðQj;0

nÞj > 0. Since jAj ¼ d2

n1₂

2 e > 2n 5 jFðQ j;0

nÞj for n P 4, there exists a link ðw; bÞ of A such that

ðw; ðwÞjÞ; ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. Next, suppose that jFðQj;0nÞj ¼ 0 and n P 5. Since jAj ¼ d2

n1

2

2 e > 2n 5, there

still exists a link ðw; bÞ of A such that ðw; ðwÞjÞ; ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. Finally, suppose that jFðQj;0_nÞj ¼ 0 and n ¼ 4. If there does not exist any node z of V1ðQj;04Þ such that ðz; ðzÞ

j

Þ is faulty, there must exist a link ðw; bÞ on H0such that

ðw; ðwÞjÞ, ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. If there exists a node z of V1ðQj;04Þ such that ðz; ðzÞ j

Þ is faulty, then it follows fromTheorem 5that Qj;0₄ fzg has a hamiltonian path, still namely H0, between u and

v

. Obviously, there also exists a link

ðw; bÞ on H0 such that ðw; ðwÞjÞ; ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. In summary, H0 can be written as

hu; H00;w; b; H 00

0;

v

i. Since jFðQ j;1

nÞj 6 n 3, it follows fromTheorem 6that Q j;1 n FðQ

j;1

nÞ contains a path H1of odd length l1

be-tween ðwÞj_{and ðbÞ}j_{for any odd integer l}

1satisfying 1 6 l162n1 1. As a result, hu; H00;w; ðwÞ j

;H1;ðbÞj;b; H000;

v

i is a path of

even length in the range from 2n1_{to 2}n

2.

The paths of lengths less than 2n1 are obtained as follows. ByCorollary 2, we have d¼ dQnFðu;

v

Þ 6 hðu;

v

Þ þ 4 and

d_Qj;0

nFðQj;0nÞðu;

v

Þ 6 hðu;

v

Þ þ 4. By inductive hypothesis, Q

j;0 n FðQ

j;0

nÞ has a path T0of length l0between u and

v

for any even

length from d_Qj;0

nFðQj;0nÞðu;

v

Þ to 2

n1

2. If d¼ hðu;

v

Þ or d¼ hðu;

v

nFðQj;0nÞðu;

v

Þ ¼ d . If d¼ hðu;

v

Þ þ 2, then d_Qj;0 nFðQ j;0 nÞðu;

v

Þ 6 d þ 2.

_v

are in Qj;1

n. Since jFðQ j;1

nÞj 6 n 3, it follows fromLemma 1that d ₆_hðu;

v

Þ þ 2. Thus, Qn F has a shortest path between u and

v

that does not cross the dimension j. By the inductive hypothesis,

Qj;1n FðQ j;1

(12)

a path of length 2n1

2 between u and

v

in Qj;1 n FðQ

j;1

nÞ. Moreover, let A ¼ fðT1ðiÞ; T1ði þ 1ÞÞ j 1 6 i 6 2n1 1;

i 1 ðmod 2Þg be a set of disjoint links on T1. First, suppose that jFðQj;1nÞj > 0. Since jAj ¼ d2

n1

2

2 e > 2n 5 jFðQ j;1 nÞj for

n P 4, there exists a link ðw; bÞ 2 A such that ðw; ðwÞj

Þ, ðb; ðbÞjÞ, and ððwÞj;ðbÞjÞ are all fault-free. Next, suppose that jFðQj;1nÞj ¼ 0 and n P 5. Since jAj ¼ d2

n1₂

2 e > 2n 5, there still exists a link ðw; bÞ 2 A such that ðw; ðwÞ j

Þ; ðb; ðbÞjÞ and ððwÞj;ðbÞjÞ are all fault-free. Finally, suppose that jFðQj;1nÞj ¼ 0 and n ¼ 4. If there does not exist any node z of V1ðQj;14Þ such

that ðz; ðzÞjÞ is faulty, there exists a link ðw; bÞ on T1such that ðw; ðwÞjÞ, ðb; ðbÞjÞ and ððwÞj;ðbÞjÞ are all fault-free. If there exists

a node z of V1ðQj;14Þ such that ðz; ðzÞ j

Þ is faulty,Theorem 5ensures that Qj;14 fzg has a hamiltonian path, still namely T1,

be-tween u and

v

. Obviously, there also exists a link ðw; bÞ on T1such that ðw; ðwÞjÞ; ðb; ðbÞjÞ and ððwÞj;ðbÞjÞ are all fault-free. In

summary, T1can be written as hu; T01;w; b; T 00

1;

v

i. Since jFðQ j;0

nÞj 6 2n 7, it follows fromTheorem 8that Q j;0 n FðQ

j;0

nÞ contains

a path T0 of length l0 between ðwÞj and ðbÞj for any odd integer l0 from 1 to 2n1 1 excluding 3. As a result,

hu; T0 1;w; ðwÞ

j

;T0;ðbÞj;b; T001;

v

i is a path of any even length in the range from 2 n1

to 2n 2, excluding 2n1þ 2.

The path of length 2n1þ 2 is discussed as follows. When n ¼ 4, jFðQj;0nÞj 6 1. Thus, there exists an integer k of

f0; 1; 2; 3g fj; dimððw; bÞÞg such that ððwÞj;ððwÞjÞkÞ, ððbÞj;ððbÞjÞkÞ, and ðððwÞjÞk;ððbÞjÞkÞ are all fault-free. Hence, hu; T01;w; ðwÞ

j_;

ððwÞjÞk;ððbÞjÞk, ðbÞj_;_{b; T}00

1;

v

i is a path of length 10. When n P 5, we have jAj jFj ¼ d2

n1₂

2 e ð2n 5Þ P 2. Thus,

there is another link ðx; yÞ of A, other than ðw; bÞ, such that ðx; ðxÞjÞ; ðy; ðyÞjÞ, and ððxÞj;ðyÞjÞ are all fault-free. Without loss of generality, T1can be written as hu; R01;w; b; R001;x; y; R0001;

v

i. Hence, hu; R01;w; ðwÞ

j ;ðbÞj;b; R00 1;x; ðxÞ j , ðyÞj;R000 1;

v

i is a path of length 2n1 þ 2.

Subcase II.1.3: Suppose that u is in Qj;0_n and

v

is in Qj;1_n. ByTheorem 2, there exists a shortest path P_{between u and}

_v

_in

Qn F such that Pcrosses the dimension j exactly once. Thus, Pcan be written as hu; P0;x; ðxÞj;P1;

v

i, where P0is a shortest

path joining u to some node x in Qj;0 n FðQ

j;0

nÞ and P1is a shortest path joining ðxÞjto

v

in Qj;1n FðQ j;1 nÞ.

Subcase II.1.3.1: Suppose that ‘ðP0Þ > 0 and ‘ðP1Þ > 0. ByTheorem 6, Qj;1n FðQ j;1

nÞ has a path T1of length l1between ðxÞj

and

v

for each l1satisfying ‘ðP1Þ 6 l162n1 1 and 2jðl1 ‘ðP1ÞÞ. Suppose that ‘ðP0Þ ¼ 1. ByTheorem 8, Qj;0n FðQ j;0 nÞ has a

path T0of length l0between u and x for any odd integer l0from 1 to 2n1 1 excluding 3. Suppose that ‘ðP0Þ > 1. By the

inductive hypothesis, Qj;0n FðQ j;0

nÞ has a path T0of length l0between u and x for each l0satisfying ‘ðP0Þ 6 l062n1 1

and 2jðl0 ‘ðP0ÞÞ. Hence, hu; T0;x; ðxÞj;T1;

v

i is a path of even length l0þ l1þ 1 in the range from dto 2n 2.

Subcase II.1.3.2: Suppose that ‘ðP0Þ ¼ 0 or ‘ðP1Þ ¼ 0. With symmetry, we assume u ¼ x. By the inductive hypothesis,

Qj;1_n FðQj;1_nÞ contains a path T1 of length l1 between ðuÞj and

v

for any odd integer l1 form ‘ðP1Þ to 2n1 1. Then

hu; ðuÞj;T1;

v

i is a path of even length l1þ 1 in the range from ‘ðP1Þ þ 1 ¼ dto 2n1.

The paths of lengths greater than 2n1_{are constructed as follows. Since jVðQ}j;0

nÞ fugj ð2n 5Þ > 1 for n P 4, we can

choose a node y from VðQj;0_nÞ fug such that ðy; ðyÞjÞ is fault-free and ðyÞj is not

v

. Let R0 be a path joining u to y in

Qj;0_n FðQj;0_nÞ and R1be a path joining ðyÞjto

v

in Qj;1n FðQ j;1

nÞ. Similar to Subcase II.1.3.1, H ¼ hu; R0;y; ðyÞj;R1;

v

i is a path

of even length in the range from d0¼ d_Qj;0 nFðQ j;0 nÞðu; yÞ þ dQ j;1 nFðQ j;1 nÞððyÞ j ;

v

Þ þ 1 to 2n 2. By Corollary 3, we have d06_{ðn þ 1Þ þ ðn 1Þ þ 1 6 2}n1_{þ 2 for n P 4. Therefore, H is a path of even length in the range from 2}n1_{þ 2 to 2}n_2.

Subcase II.2: Suppose that jFðQj;0nÞj 6 2n 6 or jFðQ j;1

jFðQj;0nÞj ¼ 2n 6. Thus, Q j;1

n is fault-free. It is noticed that the faulty links are distributed as shown inFig. 2.

v

are in Qj;0

n. Let ðw; bÞ be a faulty link of Q j;0

n such that both ðw; ðwÞ j

Þ and ðb; ðbÞjÞ are fault-free. Let F0¼ FðQj;0nÞ fðw; bÞg. By the inductive hypothesis, Q

j;0

n F0has a path Plof length l between u and

v

for

any even integer l from d_Qj;0

nF0ðu;

v

Þ to 2 n1 2. If ðw; bÞ is on Pl, we write Pl as hu; P0l;w; b; P 00 l;

v

i and deﬁne ePl¼ hu; P0l;w; ðwÞ j ;ðbÞj;b; P00

l;

v

i. Otherwise, Plcan be written as hu; P0l;x; y; P 00

l;

v

i, where ðx; yÞ is a link on Plsuch that both

ðx; ðxÞjÞ and ðy; ðyÞjÞ are fault-free. Similarly, we deﬁne ePl¼ hu; P0l;x; ðxÞ j

;ðyÞj;y; P00

l;

v

i. Then ePlis a path of length l þ 2. By

Cor-ollary 2, we have d

¼ dQnFðu;

v

Þ 6 hðu;

v

Þ þ 4 and dQj;0

nF0ðu;

v

Þ 6 hðu;

v

Þ þ 4. If dQj;0nF0ðu;

v

Þ ¼ d

_{, then path e}_P

lis the desired

path. Otherwise, if d_Qj;0

nF0ðu;

v

Þ ¼ d

þ 2, then ePlis a path of even length in the range from dþ 4 to 2n1. It is noticed that a

shortest path between u and

v

in Qn F can be constructed based on a breadth-ﬁrst search.

ByTheorem 6, Qj;1

n contains a path T1of length l1between ðwÞjand ðbÞjor a path R1of odd length l1between ðxÞjand ðyÞj

for any odd integer l1from 1 to 2n1 1. Thus, hu; P02n1₂;w; ðwÞ

j

;T1;ðbÞj;b; P002n1₂;

v

i (or hu; P0₂n1₂;x; ðxÞ

j

;R1;ðyÞj;y; P002n1₂;

v

i)

is a path of even length in the range from 2n1 to 2n 2. Subcase II.2.2: Suppose that both u and

_v

are in Qj;1

n. Let ðw; ðwÞ i

Þ be a faulty link of Qj;0n such that both ðw; ðwÞ j

Þ and ððwÞi;ððwÞiÞjÞ are fault-free. Since n P 4, we assume that t 2 f0; 1; . . . ; n 1g fj; ig. Moreover, we assume that w 2 V0ðQj;0nÞ. Let X ¼ fððwÞ

j

;ððwÞjÞkÞ j k R fi; j; tgg. Since jXj ¼ n 3, our inductive hypothesis ensures that Qj;1n X contains

a path T1 of even length l1 between u and

v

for d6l162n1 2. Let T1 denote the longest path between u and

v

in

Qj;1

n X. It is noted that ððwÞ j_;

ððwÞjÞiÞ is on T1. Hence, T1 can be represented as hu; T01;ðwÞ j_;

ððwÞjÞi;T00

1;

v

i. By the inductive

hypothesis, Qj;0_n ðFðQj;0_nÞ fðw; ðwÞiÞgÞ contains a path T0of odd length l0between w to ðwÞifor 5 6 l062n1 1. As a result,

hu; T0 1;ðwÞ

j

;w; T0;ðwÞi;ððwÞjÞi;T001;

v

i is a path of even length 2 n1

þ l0 1, in the range from 2n1þ 4 to 2n 2.

Let A ¼ fðT1ðkÞ; T1ðk þ 1ÞÞ j 1 6 k 6 2n1 1; k 1 ðmod 2Þg be a set of disjoint links on T1. Then the paths of lengths 2n1

and 2n1_{þ 2 can be obtained as follows. When n ¼ 4, we suppose that fp; q; j; ig ¼ f0; 1; 2; 3g. Since ðw; ðwÞ}i

Þ is faulty, we have either fðw; ðwÞpÞ; ððwÞp;ððwÞpÞiÞ; ððwÞpÞi;ðwÞiÞg \ F ¼ ; or fðw; ðwÞqÞ; ððwÞq;ððwÞqÞiÞ; ððwÞqÞi, ðwÞi;ðwÞqÞiÞg \ F ¼ ;. Without loss