The spanning laceability on the faulty bipartite hypercube-like networks

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The spanning laceability on the faulty bipartite hypercube-like

networks

Cheng-Kuan Lin

a

, Yuan-Hsiang Teng

b,⇑

, Jimmy J.M. Tan

c

, Lih-Hsing Hsu

d

, Dragan Marušicˇ

e a

Institute of Information Science, Academia Sinica, Taipei City 11529, Taiwan, ROC

b

Department of Computer Science and Information Engineering, Hungkuang University, Taichung City 433, Taiwan, ROC

c

Department of Computer Science, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC

d

Department of Computer Science and Information Engineering, Providence University, Taichung City 433, Taiwan, ROC

e

IMFM, University of Ljubljana, 1111 Ljubljana, Slovenia

a r t i c l e

i n f o

Keywords: Hamiltonian Hamiltonian laceable Hypercube networks Hypercube-like network Spanning laceability

a b s t r a c t

A w-container Cðu;

v

Þ of a graph G is a set of w-disjoint paths joining u to

v

. A w-container of G is a w_{-container if it contains all the nodes of VðGÞ. A bipartite graph G is w}_{-laceable if} there exists a w_{-container between any two nodes from different parts of G. Let n and k be} any two positive integers with n P 2 and k 6 n. In this paper, we prove that n-dimensional bipartite hypercube-like graphs are f-edge fault k-laceable for every f 6 n 2 and f þ k 6 n.

1. Introduction

1.1. Basic graph deﬁnitions and notations

The research about interconnection networks is important for parallel and distributed computer systems. The layouts of processors and links in distributed computer systems are usually represented by a network structure. Computer network topologies are usually represented by graphs where nodes represent processors and edges represent links between proces-sors. The containers of graphs do exist in information engineering design, telecommunication networks, and biological neu-ral systems ([1,2]and its references). The study of w-container, w-wide distance, and their w_{-versions play a pivotal role in}

the design and the implementation of parallel routing and efﬁcient information transmission in large scale networking sys-tems. In bioinformatics and neuroinformatics, the existence as well as the structure of a w_{-container signiﬁes the cascade}

effect in the signal transduction system and the reaction in a metabolic pathway.

For graph deﬁnitions and notations, we follow[3,4]. Let G ¼ ðV; EÞ be a graph where V is a ﬁnite set and E is a subset of fðu;

v

Þjðu;

v

Þ is an unordered pair of Vg. We say that V is the node set and E is the edge set. We use nðGÞ to denote jVj. Two nodes u and

v

are adjacent if ðu;

v

Þ 2 E. For a node u, we use NGðuÞ to denote the neighborhood of u which is the set

f

v

jðu;

v

Þ 2 Eg. For any node u of V, we denote the degree of u by degGðuÞ ¼ jNGðuÞj. A graph G is k-regular if degGðuÞ ¼ k for

every node u in G. A path P between nodes

v

1and

v

kis a sequence of adjacent nodes, h

v

1;

v

2; . . . ;

v

ki, in which the nodes

v

1;

v

2; . . . ;

v

k are distinct except that possibly

v

1¼

v

k. We use P1 to denote the path h

v

k;

v

k1; . . . ;

v

1i. The length of

P; lðPÞ, is the number of edges in P. We also write the path P as h

v

1;

v

2; . . . ;

v

i;Q ;

v

j;

v

jþ1; . . . ;

v

ki, where Q is the path

h

v

i;

v

iþ1; . . . ;

v

ji. Hence, it is possible to write a path as h

v

1;

v

2;Q ;

v

2;

v

3; . . . ;

v

ki if lðQ Þ ¼ 0. Let IðPÞ ¼ VðPÞ f

v

1;

v

kg be the

set of the internal nodes of P. A set of paths fP1;P2; . . . ;Pkg are internally node-disjoint (abbreviated as disjoint) if

http://dx.doi.org/10.1016/j.amc.2013.02.027

⇑Corresponding author.

E-mail address:yhteng@sunrise.hk.edu.tw(Y.-H. Teng).

Contents lists available atSciVerse ScienceDirect

Applied Mathematics and Computation

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IðPiÞ \ IðPjÞ ¼ ; for any i – j. A path is a hamiltonian path if it contains all nodes of G. A graph G is hamiltonian connected if

there exists a hamiltonian path joining any two distinct nodes of G[5]. A cycle is a path with at least three nodes such that the ﬁrst node is the same as the last one. A hamiltonian cycle of G is a cycle that traverses every node of G. A graph is ham-iltonian if it has a hamham-iltonian cycle. A graph G is bipartite if its node set can be partitioned into two subsets V1and V2such

that every edge connects nodes between V1and V2. A bipartite graph G is hamiltonian laceable if there is a hamiltonian path

of G joining any two nodes from distinct bipartition[6]. A bipartite graph G is k-edge fault hamiltonian laceable if G F is ham-iltonian laceable for any edge subset F of G with jFj 6 k.

A graph G is k-connected if there exists a set of k internally disjoint paths fP1;P2; . . . ;Pkg between any two distinct nodes u

and

v

. A subset S of VðGÞ is a cut set if G S is disconnected. A container Cðu;

v

Þ between two distinct nodes u and

v

in G is a set of disjoint paths between u and

v

. A w-container Cwðu;

v

Þ in a graph G is a set of w internally node-disjoint paths between u

and

v

. The concepts of a container and of a wide distance were proposed by Hsu[2]to evaluate the performance of commu-nication for an interconnection network. The connectivity of G,

j

ðGÞ, is the minimum number of nodes whose removal leaves the remaining graph disconnected or trivial. Hence, a graph G is k-connected if

j

ðGÞ P k. It follows from Menger’s Theorem

[7]that there is a w-container for w 6 k between any two distinct nodes of G if G is k-connected. 1.2. w_{-connected graphs and w}_{-laceable graphs}

In this paper, we are interested in a speciﬁc type of container. A w_{-container C}

wðu;

v

Þ in a graph G is a w-container such

that every node of G is on some path in Cwðu;

v

Þ. A graph G is w-connected if there exists a w-container between any two

distinct nodes in G. Obviously, we have the following remark.

Remark 1. ð1:aÞ a graph G is 1-connected if and only if it is hamiltonian connected[5], ð1:bÞ a graph G is 2-connected if it is hamiltonian, and ð1:cÞ an 1-connected graph except K1and K2is 2-connected.

The study of w_{-connected graph is motivated by the 3}_{-connected graphs proposed by Albert et al.}_[8]_{. Some related}

works have appeared in[8,9]. Assume that the graph G is w_{-connected with w 6}

_j

_{ðGÞ. The spanning connectivity of a graph}

G;

j

_{ðGÞ, is the largest integer k such that G is i}_{-connected for every i with 1 6 i 6 k. A graph G is super spanning connected if}

j

_{ðGÞ ¼}

_j

_{ðGÞ. In such case, the number}

_j

_{ðGÞ ¼}

_j

_{ðGÞ is called the super spanning connectivity of G. In}_[10–13]_{, some families}

of graphs are proved to be super spanning connected.

A bipartite graph is said to be w_{-laceable if there exists a w}_{-container between any two nodes from different partite sets}

for some w with 1 6 w 6

j

ðGÞ. Any bipartite w_{-laceable graph with w P 2 has the equal size of bipartition. We have the}

following remark.

Remark 2. ð2:aÞ an 1_{-laceable graph is also known as hamiltonian laceable graph}_[6]_{, ð2:bÞ a graph G is 2}_{-laceable if and} only if it is hamiltonian, and ð2:cÞ an 1-laceable graph except K1and K2are 2-laceable.

The spanning laceability of a bipartite graph G;

j

L_{ðGÞ, is the largest integer k such that G is i}_{-laceable for every i with}

1 6 i 6 k. A graph G is super spanning laceable if

j

L_{ðGÞ ¼}

j

_{ðGÞ. Recently, Chang et al.}_[14]_{proved that the n-dimensional}

hypercube Qnis super spanning laceable for every positive integer n. It was proved in[11]that the n-dimensional star graph

Snis super spanning laceable if and only if n – 3.

1.3. Hypercube-like graphs H0n

Among all interconnection networks proposed in the literature, the hypercube Qnis one of the most popular topologies

[14–17]. However, the hypercube does not have the smallest diameter for its resources. Various networks are proposed by twisting some pairs of links in hypercubes[18–21]. Because of the lack of the uniﬁed perspective on these variants, results of one topology are hard to be extended to others. To make a uniﬁed study of these variants, Vaidya et al. introduced the class of hypercube-like graphs[22]. We denote these graphs as H0_{-graphs. The class of H}0_{-graphs, consisting of simple, connected,}

and undirected graphs, contains most of the hypercube variants.

Let G0¼ ðV0;E0Þ and G1¼ ðV1;E1Þ be two disjoint graphs with the same number of nodes. A 1–1 connection between G0

and G1 is deﬁned as E ¼ fð

v

;/ð

v

ÞÞj

v

2 V0;/ð

v

Þ 2 V1, and / : V0! V1 is a bijectiong. We use G0 G1 to denote

G ¼ ðV0[ V1;E0[ E1[ EÞ. The operation ‘‘’’ may generate different graphs depending on the bijection /. There are some

studies on the operation ‘‘’’[23,24]. Let G ¼ G0 G1, and let x be any node in G. We use x to denote the unique node

matched under /.

Now, we can deﬁne the set of n-dimensional H0-graph, H0n, as follows:

(1) H0

1¼ fK2g, where K2is the complete graph with two nodes.

(2) Assume that G0;G12 H0n. Then G ¼ G0 G1is a graph in H0nþ1.

We can deﬁne the set of bipartite n-dimensional H0_{-graph, B}0

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(1) B0

1¼ fK2g, where K2is the complete graph deﬁned on fa; bg with bipartition V0¼ fag and V1¼ fbg.

(2) For i ¼ 0; 1, let Gibe a graph in B0nwith bipartition V i 0and V

i

1. Let / be a bijection between V 0 0[ V 0 1and V 1 0[ V 1 1such that /ð

v

Þ 2 V11iif

v

2 V 0 i. Then G ¼ G0 G1is a graph in B0nþ1. Every graph in H0

nis an n-regular graph with 2

n_{nodes, and every graph in B}0

ncontains 2

n1_{nodes in each bipartition. Note}

that the n-dimensional hypercube Qn2 B0n.

Let G be a graph in H0nþ1. Then G ¼ G0 G1with both G0and G1in H0n. Let u be a node in VðGÞ. Then u is a node in VðGiÞ for

some i ¼ 0; 1. We use u to denote the node in VðG1iÞ matched under /. So u ¼

v

if u ¼

v

.

In the following section, we give some properties about the bipartite n-dimensional hypercube-like graphs B0

n. Let n and k

be any two positive integers with n P 2 and k 6 n. In Section3and Section4, we prove that every B0

nis f-edge fault k

-lace-able for every f 6 n 2 and f þ k 6 n. We give our conclusion in the ﬁnal section.

2. Preliminaries

Park and Chwa[25]studied the hamiltonian laceability properties of the bipartite hypercube-like networks. Some results are listed as follows.

Theorem 1 [25]. Every graph in B0

nis hamiltonian laceable, and every graph in B0nis hamiltonian if n P 2. Theorem 2 [25]. Suppose that n P 2; i 2 f0; 1g, and G is a graph in B0

nwith bipartition G0and G1. Let fu1;u2g # VðGiÞ with

u1–u2, and f

v

1;

v

2g # VðG1iÞ with

v

1–

v

2. Then there are two disjoint paths P1and P2of G such that (1) P1joins u1to

v

1,

(2) P2joins u2to

v

2, and (3) P1[ P2spans G.

The fault-tolerance hamiltonian laceability of the bipartite hypercube-like networks is studied by Lin et al. in[26]. Theorem 3 [26]. Let n P 2. Every graph in B0

nis ðn 2Þ-edge fault hamiltonian laceable. Theorem 4 [26]. Suppose that n P 2; i 2 f0; 1g, and G is a graph in B0

n with bipartition G0 and G1. Let z 2 VðGiÞ, and

fu;

v

g # VðG1iÞ with u –

v

. Then there is a hamiltonian path of G fzg joining u to v.

3. The super spanning laceability of the graph in B0 n

Let n and k be any two positive integers with n P 2 and k 6 n. In this section, we show that every graph in B0

nis f-edge

fault k_{-laceable for every f 6 n 2 and f þ k 6 n. We give the concept of the spanning fan ﬁrst. We note that there is another}

Menger-type Theorem. Let u be a node of G and S ¼ f

v

1;

v

2; . . . ;

v

kg be a subset of VðGÞ not including u. An ðu; SÞ-fan is a set of

disjoint paths fP1;P2; . . . ;Pkg of G such that Pijoins u to

v

ifor every 1 6 i 6 k[27]. It is proved that a graph G is k-connected if

and only if there exists an ðu; SÞ-fan between any node u and any k-subset S of VðGÞ such that u R S. With this observation, we deﬁne a spanning fan is a fan that spans a graph G. Naturally, we can study

j

fanðGÞ as the largest integer k such that there

exists a spanning ðu; SÞ-fan between any node u and any k-node subset S with u R S. However, we defer such a study for the following reasons.

First, let S be a cut set of a graph G. Let u be any node of VðGÞ S. It is easy to see that there is no spanning ðu; SÞ-fan in G. Thus,

j

fanðGÞ <

j

ðGÞ if G is not a complete graph.

Second, let G be a bipartite graph with bipartition G0¼ ðV0;E0Þ and G1¼ ðV1;E1Þ such that jV0j ¼ jV1j. Let u be a node in Vi

with i 2 f0; 1g; S ¼ f

v

1;

v

2; . . . ;

v

kg # VðGÞ fug, and k 6

j

ðGÞ. Suppose that jS \ V1ij ¼ r. Without loss of generality, we

as-sume that f

v

1;

v

2; . . . ;

v

rg V1i. Let fP1;P2; . . . ;Pkg be any spanning ðu; SÞ-fan of G. Then lðPiÞ is odd if i 6 r, and lðPiÞ is even if

r < i 6 k. Let lðPiÞ ¼ 2tiþ 1 if i 6 r and lðPiÞ ¼ 2tiif i > r. For i 6 r, there are ti 1 nodes of Piin Viother than u, and there are ti

nodes of Pi in V1i. For i > r, there are tinodes of Piin Viother than u, and there are tinodes of Pi in V1i. Thus, we have

jVij ¼ 1 r þPki¼1tiand jV1ij ¼Pki¼1ti. Since jVij ¼ jV1ij; r ¼ 1. Thus, r ¼ 1 is a natural requirement as we study the

span-ning fan of bipartite graphs with equal size of bipartition.

Theorem 5 [12]. Suppose that n and k are two positive integers with k 6 n. Let G be a graph in B0

nwith bipartition G0and G1. There exists a spanning ðu; SÞ-fan in G for any node u in VðGiÞ and any node subset S with jSj ¼ k 6 n such that u R S, and jS \ VðG1iÞj ¼ 1 with i 2 f0; 1g.

Lemma 1. Suppose that

1. n P 2; f ¼ n 2, and i 2 f0; 1g,

2. G is a graph in B0nwith bipartition G0and G1, and

3. F EðGÞ with jFj ¼ f .

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Proof. ByTheorem 3, there is a hamiltonian path P ¼ hx; P1;u; P2;yi of G F joining x to y. Then fP1;P2g is the spanning

ðu; fx; ygÞ-fan of G F. h The following are the main results. Theorem 6. Suppose that

1. n P 2; k 6 n, and i 2 f0; 1g,

2. G is a graph in B0nwith bipartition G0¼ ðV0;E0Þ and G1¼ ðV1;E1Þ, and

3. F EðGÞ with jFj þ k 6 n and jFj 6 n 2.

Then, for any u 2 Viand S # VðGÞ fug with jSj ¼ k and jS \ V1ij ¼ 1, there exists a spanning ðu; SÞ-fan in G F.

We prove the theorem by induction. However, the proof of the theorem is rather long. We prove it in the following section.

Theorem 7. The bipartite n-dimensional hypercube-like graph B0

nis f-edge fault k

-laceable for f 6 n 2 and f þ k 6 n. Proof. Let G be a graph in B0

nwith bipartition G0and G1. Assume that x 2 VðGiÞ and y 2 VðG1iÞ for some i 2 f0; 1g. Suppose

that F EðGÞ with jFj ¼ f and f 6 n 2. Let S # VðGiÞ fxg adjacent to y in G F with jSj ¼ k 1 and k 6 n f . We assume

that S ¼ fy1;y2; . . . ;yk1g. ByTheorem 6, there exists a spanning ðx; S [ fygÞ-fan fP1;P2; . . . ;Pkg in G F such that Pkjoins x to

y, and Pijoins x to yifor 1 6 i 6 k 1. Let Qi¼ hx; Pi;yi;yi for 1 6 i 6 k 1. Thus, fPk;Q1;Q2; . . . ;Qk1g forms a k-container

between x and y in G F. The theorem is proved. h

4. Proof of Theorem 6

Let G ¼ G0 G1in B0nwith bipartition V j 0and V j 1for j 2 f0; 1g. Thus, V 0 0[ V 1 0and V 0 1[ V 1

1form the bipartition of G. Assume

that jFj ¼ f . Let u be any node in V00[ V 1

0and S ¼ f

v

1;

v

2; . . . ;

v

kg be any node subset in G fug with

v

1being the unique node

in ðV01[ V 1

1Þ \ S. Without loss of generality, we assume that u 2 V 0

0. For n ¼ 2, we have G is isomorphic to a cycle with four

nodes. Thus, this statement holds on n ¼ 2. ByLemma 1, Theorem 3, and Theorem 5, this statement holds on n ¼ 3. Thus, we assume that n P 4. ByLemma 1 and Theorem 3, this statement holds on k 2 f1; 2g and f ¼ n 2. ByTheorem 5, this statement holds on k 6 n and f ¼ 0. Thus, we assume that k P 3 and 1 6 f 6 n 3 with k þ f 6 n. We set T ¼ S f

v

1g; Fj¼ F \ EðGjÞ for j 2 f0; 1g, and F2¼ F ðF0[ F1Þ. Note that jFj ¼ jF0j þ jF1j þ jF2j and jFj0j 6 n 3 for every

j0 2 f0; 1; 2g. Now we have the following cases. Case 1. jT \ V0 0j ¼ jTj.

v

1

v

2

u

v

3

v

4

v

5

v

6

x

G

1

v

2

v

3

u

v

4

v

5

v

1

v

6

x

G

1

u

x

u

v

2

v

3

u

v

4

v

5

v

1

v

6

x

G

y

1

y

u

x

(a)

(b)

(c)

(d)

v

2

v

3

u

v

4

v

5

v

1

v

6

x

y

_y

x

G

1

G

₀

- F

₀

G

₀

- F

₀

G

₀

- F

₀

G

₀

- F

₀

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Case 1.1. jF0j ¼ jFj and

v

12 V01. Let H ¼ S f

v

kg. We have H G0;jH \ V01j ¼ 1, and jHj ¼ k 1. By induction, there is a

span-ning ðu; HÞ-fan fP1;P2; . . . ;Pk1g of G0 F0. Without loss of generality, we assume that Pijoins u to

v

ifor every 1 6 i 6 k 1.

Suppose that

v

k2 VðP1Þ. Without loss of generality, we write P1 as hu; Q1;

v

k;x; Q2;

v

1i. Since

v

k2 V00;x 2 V 0

1. (Note that

x ¼

v

1 if lðQ2Þ ¼ 0.) By Theorem 1, there is a hamiltonian path R of G1 joining node u 2 V11 to node x 2 V 1

0. We set

W1¼ hu; u; R; x; x; Q2;

v

1i; Wi¼ Pifor every 2 6 i 6 k 1, and Wk¼ hu; Q1;

v

ki. Then fW1;W2; . . . ;Wkg forms the spanning

ðu; SÞ-fan of G F. SeeFig. 1(a) for an illustration. Suppose that

v

k2 VðPiÞ for some 2 6 i 6 k 1. Without loss of generality,

we assume that

v

k2 VðPk1Þ and write Pk1as hu; Q1;

v

k;x; Q2;

v

k1i. Since

v

k2 V00;x 2 V 0

1. ByTheorem 1, there is a

hamilto-nian path R of G1joining node u 2 V11to node x 2 V 1

0. We set Wi¼ Pifor every 1 6 i 6 k 2; Wk1¼ hu; u; R; x; x; Q2;

v

k1i, and

Wk¼ hu; Q1;

v

ki. Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F. SeeFig. 1(b) for an illustration.

v

12 V11. We choose a node x in V01. Let H ¼ ðT [ fxgÞ f

v

kg. We have H G0;jH \ V01j ¼ 1, and jHj ¼ k 1. By induction, there is a spanning ðu; HÞ-fan fP1;P2; . . . ;Pk1g of G0 F0such that P1joins u to x, and Pijoins u to

v

i for every 2 6 i 6 k 1. Note that u 2 V11 and x 2 V 1

0. Without loss of generality, we assume that

v

k2 VðP1Þ. Let P1¼ hu; Q1;y;

v

k;Q2;xi. Since

v

k2 V00, we have y 2 V01and y 2 V10.

Suppose that

v

1– u. ByTheorem 2, there are two disjoint paths R1and R2in G1such that ð1Þ R1joins y to

v

1;ð2Þ R2joins u

to x, and ð3Þ R1[ R2spans G1. We set W1¼ hu; Q1;y; y; R1;

v

1i; Wi¼ Pifor every 2 6 i 6 k 1, and Wk¼ hu; u; R2; x; x; Q12 ;

v

ki.

Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F. SeeFig. 1(c) for an illustration.

Suppose that

v

1¼ u. By Theorem 4, there is a hamiltonian path R of G1 f

v

1g joining y to x. We set

W1¼ hu; u ¼

v

1i; Wi¼ Pifor every 2 6 i 6 k 1, and Wk¼ hu; Q1;y; y; R; x; x; Q12 ;

v

ki. Then fW1;W2; . . . ;Wkg forms the

span-ning ðu; SÞ-fan of G F. SeeFig. 1(d) for an illustration.

Case 1.3. jF0j < jFj and

v

12 V01. Since jF0j < jFj ¼ f , we have k þ jF0j 6 k þ f 1 6 n 1. By induction, there is a spanning ðu; SÞ-fan fP1;P2; . . . ;Pkg of G0 F0. Without loss of generality, we assume that Pi joins u to

v

i for every 1 6 i 6 k. Since jVðG0Þj ¼ 2n1and [k

i¼1Pispan G0, we have Pk

i¼1jEðPiÞj ¼ 2n1 1. Since 2n1 1 > 3n 8 > 2ðf 1Þ þ k if n P 3, there exists an edge ðx; yÞ in [k

i¼1EðPiÞ such that ðx; xÞ R F2and ðy; yÞ R F2. Without loss of generality, we assume that ðx; yÞ 2 EðPjÞ for some 1 6 j 6 k. Let Pj¼ hu; R1;x; y; R2;

v

ji. Note that u ¼ x if lðR1Þ ¼ 0 and y ¼

v

jif lðR2Þ ¼ 0. Since x and y are adjacent, x and y are in distinct bipartition of G0. Moreover, x and y are in distinct bipartition of G1. ByTheorem 3, there is a hamiltonian path W of G1 F1 joining x to y. We set Wi¼ Pi for every i 2 f1; 2; . . . ; kg fjg and set Wj¼ hu; R1;x; x; W; y; y; R2;

v

ji. Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F.

v

12 V11. Since jV 0 1j ¼ 2

n2_>_{n 3 if n P 3, there exists a node x 2 V}0

1 such that ðx; xÞ R F2. Let

H ¼ T [ fxg. Since jF0j < jFj ¼ f , we have k þ jF0j 6 k þ f 1 6 n 1. By induction, there is a spanning ðu; HÞ-fan

fP1;P2; . . . ;Pkg of G0 F0. Without loss of generality, we assume that P1is joining u to x and Piis joining u to

v

ifor every

2 6 i 6 k. Since x 2 V0

1, we have x 2 V 1

0. By Theorem 3, there is a hamiltonian path R of G1 F1joining x to

v

1. We set

W1¼ hu; P1;x; x; R;

v

1i and Wi¼ Pifor every 2 6 i 6 k. Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F.

Case 2. jT \ V10j ¼ 1. We assume that

v

k2 V10. Note that u 2 V 1 1.

v

12 V01. Let H ¼ S f

v

kg. We have H G0;jH \ V01j ¼ 1, and jHj ¼ k 1. By induction, there is a

span-ning ðu; HÞ-fan fW1;W2; . . . ;Wk1g of G0 F0. ByTheorem 1, there is a hamiltonian path R of G1joining u to

v

k. We set

Wk¼ hu; u; R;

v

ki. Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F. SeeFig. 2(a) for an illustration.

v

12 V11. By Theorem 1, there is a hamiltonian path R of G1 joining

v

1 to

v

k. We write R as

h

v

1;R1; u; x; R2;

v

ki. Note that

v

1¼ u if lðR1Þ ¼ 0 and x ¼

v

k if lðR2Þ ¼ 0. Since u 2 V11, we have x 2 V 1

0 and x 2 V 0 1. Let

H ¼ ðT [ fxgÞ f

v

kg. Thus H G0;jH \ V10j ¼ 1, and jHj ¼ k 1. By induction, there is a spanning ðu; HÞ-fan

fP1;P2; . . . ;Pk1g of G0 F0 such that P1 joins u to x, and Pi joins u to

v

i for every 2 6 i 6 k 1. We set

v

1

v

2

u

v

3

v

4

v

5

v

6

G

v

2

v

3

u

v

4

v

5

v

6

v

1

x

G

(a)

(b)

x

u

1 1

G

₀

- F

₀

G

₀

- F

₀

(6)

W1¼ hu; u; R11 ;

v

1i; Wi¼ Pifor every 2 6 i 6 k 1, and Wk¼ hu; P1; x; x; R2;

v

ki. Then fW1;W2; . . . ;Wkg forms the ðu; SÞ-fan of

G F. SeeFig. 2(b) for an illustration. Case 2.3. jF0j < jFj and

v

12 V01. Since jV

0 0j ¼ 2

n2_>_{n > k þ f 1 if n P 4, there exists a node x in V}0

0 ðT [ fugÞ such that

ðx; xÞ R F2. Let H ¼ ðS [ fxgÞ f

v

kg. Obviously, H G0;jH \ V10j ¼ 1, and jHj ¼ k. Since jF0j < jFj ¼ f , we have

k þ jF0j 6 k þ f 1 6 n 1. By induction, there is a spanning ðu; HÞ-fan fP1;P2; . . . ;Pkg of G0 F0such that Pijoins u to

v

i

for every 1 6 i 6 k 1 and Pkjoins u to x. ByTheorem 1, there is a hamiltonian path R of G1 F1joining x to

v

k. We set

Wi¼ Pifor every 1 6 i 6 k 1 and Wk¼ hu; Pk;x; x; R;

v

ki. Then fW1;W2; . . . ;Wkg forms the spanning ðu; SÞ-fan of G F.

v

12 V11. Since jV 0 0j ¼ 2

n2_>_{n > k þ f 1 if n P 4, there exists a node x in V}0

0 ðT [ fugÞ such that

ðx; xÞ R F2. Let F0¼ fðy; xÞjy 2 G1 and ðy; yÞ 2 F2g. We have jF1[ F0j 6 jF1j þ jF2j 6 jFj ¼ f < n 3. ByTheorem 1, there is a

hamiltonian path R of G1 ðF1[ F0Þ joining

v

1to

v

k. Without loss of generality, we write R as h

v

1;R1; x; z; R2;

v

ki. Note that

v

1¼ x if lðR1Þ ¼ 0 and z ¼

v

kif lðR2Þ ¼ 0. Since x 2 V00, we have x 2 V 1 1;z 2 V 1 0, and z 2 V 0 1. Let H ¼ ðT [ fx; zgÞ f

v

kg.

Obvi-ously, H G0;jH \ V01j ¼ 1, and jHj ¼ k. Since jF0j < jFj ¼ f , we have k þ jF0j 6 k þ f 1 6 n 1. By induction, there is a

span-ning ðu; HÞ-fan fP1;P2; . . . ;Pkg of G0 F0. Without loss of generality, we assume that P1joins u to x; P2joins u to z, and Pijoins

u to

v

i1 for every 3 6 i 6 k. We set W1¼ hu; P1;x; x; R11 ;

v

1i; Wi¼ Piþ1for every 2 6 i 6 k 1, and Wk¼ hu; P2; z; z; R2;

v

ki.

Then fW1;W2; . . . ;Wkg forms the ðu; SÞ-fan of G F.

Case 3. jT \ V1

0j P 2 and jT \ V 0

0j P 1. We have n P k þ 1 ¼ jSj þ 1 P 5. Assume that A ¼ T \ V 0

0¼ f

v

2;

v

3; . . . ;

v

tg and

B ¼ T \ V1

0¼ f

v

tþ1;

v

tþ2; . . . ;

v

kg for some 2 6 t 6 k 2.

Case 3.1. jF0j ¼ jFj. Since ðn 1ÞjAj 6 ðn 1Þðn 3Þ < 2nn2if n P 5, there exists a node x in V10such that

v

1 R NG1ðxÞ and

v

i R NG1ðxÞ for 2 6 i 6 t. By induction, there is a spanning ðx; B [ fugÞ-fan fP1;P2; . . . ;Pktþ1g of G1 such that

P1¼ hx; x1;P01; ui joins x to u, and Pi¼ hx; xi;P0i;

v

tþi1i joins x to

v

tþi1for every 2 6 i 6 k t þ 1.

Case 3.1.1.

v

12 V01. We set H ¼ A [ f

v

1g [ fxij2 6 i 6 k tg. Let fQ1;Q2; . . . ;Qk1g be a spanning ðu; HÞ-fan of G0 F0such

that Qi joins u to

v

i for every 1 6 i 6 t, and Qj joins u to xjtþ1 for every t þ 1 6 j 6 k 1. We set

Wi¼ Qi;Wj¼ hu; Qj; xjtþ1;xjtþ1;P0jtþ1;

v

ji, and Wk¼ hu; u; P11 ;x; Pktþ1;

v

ki. Then fW1;W2; . . . ;Wkg forms a spanning

ðu; SÞ-fan of G F.

Case 3.1.2.

v

12 V11 and

v

12 VðP1Þ. We write P1¼ hx; R1;y;

v

1;R2; ui. We set H ¼ A [ fxij2 6 i 6 k tg [ fyg. Let

fQ1;Q2; . . . ;Qk1g be a spanning ðu; HÞ-fan of G0 F0such that Q1joins u to y; Qjjoins u to

v

jfor every 2 6 j 6 t, and Qj0joins

u to xj0tþ1 for every t þ 1 6 j0 6 k 1. We set W1¼ hu; u; R21;

v

1i; Wj¼ Qj, Wj0¼ hu; Qj0; xj0tþ1;xj0tþ1;P0j0tþ1;

v

j0i, and

Wk¼ hu; Q1; y; y; R11 ;x; Pktþ1;

v

ki. Then fW1;W2; . . . ;Wkg forms a spanning ðu; SÞ-fan of G F.

Case 3.1.3.

v

12 V11and

v

12 VðPiÞ for some 2 6 i 6 k t þ 1. Without loss of generality, we assume that

v

12 VðP2Þ. Let

P2¼ hx; R1;

v

1;y; R2;

v

tþ1i. We set H ¼ A [ fxij3 6 i 6 k t þ 1g [ fyg. Let fQ1;Q2; . . . ;Qk1g be a spanning ðu; HÞ-fan of

G0 F0such that Q1joins u to y, Qj joins u to

v

j for every 2 6 j 6 t, and Qj0 joins u to xj0tþ1 for every t þ 2 6 j0 6 k. We

set W1¼ hu; u; P11 ;x; R1;

v

1i; Wj¼ Qj, Wtþ1¼ hu; Q1; y; y; R2;

v

tþ1i, and Wj0¼ hu; Qj0; xj0tþ1;xj0tþ1;P0j0tþ1;

v

j0i. Then

fW1;W2; . . . ;Wkg forms a spanning ðu; SÞ-fan of G F.

Case 3.2. jF0j < jFj and n ¼ 5. We have jFj ¼ 1 and k ¼ 4. Thus, jF0j = 0 and jF1j þ jF2j ¼ 1. Moreover, A ¼ f

v

2g and

B ¼ f

v

3;

v

4g.

Case 3.2.1. jF1j ¼ 0 and

v

12 V01. Since jV 1

1j ¼ 8 > 3, there exist two distinct nodes x1 and x2 in V11 fu;

v

2g such that

ðx1; x1Þ R F2and ðx2; x2Þ R F2. ByTheorem 2, there are two disjoint paths P1and P2in G1such that Pijoins xi to

v

iþ2for

i 2 f1; 2g, and P1[ P2spans G1. Let fQ1;Q2;Q3;Q4g be a spanning ðu; f

v

1;

v

2; x1; x2gÞ-fan of G0such that Qijoins u to

v

ifor

1 6 i 6 2, and Qj joins u to xj2 for 3 6 j 6 4. We set Wi¼ hu; Qiþ2; xi;xi;Pi;

v

iþ2i for every 1 6 i 6 2. Then fQ1;Q2;W1;W2g

forms a spanning ðu; SÞ-fan of G F. Case 3.2.2. jF1j ¼ 0 and

v

12 V11. Since jV

1

0j ¼ 8, there exists a node x in V 1

0 f

v

3;

v

4g such that

v

2 R NG1ðxÞ and ðx; xÞ R F2.

We set F0 ¼ fðx; yÞjy 2 NG1ðxÞ and ðy; yÞ 2 F2g. We have jF0j 6 1. By induction, there is a spanning ðx; f

v

1;

v

3;

v

4gÞ-fan

fP1;P2;P3g of G1 such that P1 joins x to

v

1 and Pi joins x to

v

iþ1 for 2 6 i 6 3. We set P1¼ hx; x1;R1;

v

1i and

Pi¼ hx; xi;Ri;

v

iþ1i for every 2 6 i 6 3. Let fQ1;Q2;Q3;Q4g be a spanning ðu; f

v

2; x; x2; x3gÞ-fan of G0such that Q1joins u to

x; Q2 joins u to

v

2, and Qi joins u to xi1 for 3 6 i 6 4. We set W1¼ hu; Q1; x; x; P1;

v

1i; W2¼ Q2, and

(7)

v

12 V01. We have jF2j ¼ 0. ByTheorem 3, there is a hamiltonian path P of G1 ðF1[ fð

v

3;

v

2ÞgÞ

join-ing u to

v

4. We set P ¼ hu; P1;

v

3;x; P2;

v

4i. Let fQ1;Q2;Q3g be a spanning ðu; f

v

1;

v

2; xgÞ-fan of G0such that Qijoins u to

v

ifor

1 6 i 6 2 and Q3joins u to x. Let W1¼ hu; u; P1;

v

3i and W2¼ hu; Q3; x; x; P2;

v

4i. Then fQ1;Q2;W1;W2g forms a spanning

v

12 V11. Since jV 1

0j ¼ 8 > 6, there exists a node x in V 1

0 f

v

3;

v

4g such that

v

2 R NG1ðxÞ. By induction,

there exists a ðx; f

v

1;

v

3;

v

4gÞ-fan fP1;P2;P3g of G1 F1such that P1joins x to

v

1, and Pijoins x to

v

iþ1for 2 6 i 6 3. Without

loss of generality, we write Pi¼ hx; yi1;Ri1;

v

i1i for 2 6 i 6 3. Let fQ1;Q2;Q3;Q4g be a ðu; fx;

v

2; y1; y2gÞ-fan of G0such that

Q1 joins u to x; Q2 joins u to

v

2, and Qi joins u to yi2 for 3 6 i 6 4. We set W1¼ hu; Q1; x; x; P1;

v

1i; W2¼ Q2, and

Wi¼ hu; Qi; yi2;yi2;Ri2;

v

ii for 3 6 i 6 4. Then fW1;W2;W3;W4g forms a spanning ðu; SÞ-fan of G F.

Case 3.3. jF0j < jFj and n P 6. Since ðn 1Þðf þ jAjÞ 6 ðn 1Þðf þ k 3Þ 6 ðn 1Þðn 3Þ < 2n2if n P 6, there exists a node x

in V1

0such that ðx; xÞ R F2;

v

i R NG1ðxÞ for every 2 6 i 6 t, and ðy; yÞ R F2for every y 2 NG1ðxÞ.

Case 3.3.1.

v

12 V01. Since jAj þ f < 2 n2

if n P 6, there exists a node y in V11such that ðy; yÞ R F2and y R A.

Suppose that x R B. By induction, there is a spanning ðx; B [ fygÞ-fan fP1;P2; . . . ;Pktþ1g of G1 F1such that Pijoins x to

v

tþi

for every 1 6 i 6 k t and Pktþ1joins x to y. Without loss of generality, we set Pi¼ hx; xi;Ri;

v

tþii for every 1 6 i 6 k t 1.

We set H ¼ A [ fxij1 6 i 6 k t 1g [ fyg. Let fQ1;Q2; . . . ;Qkg be a spanning ðu; HÞ-fan of G0 F0such that Qijoins u to

v

i

for every 1 6 i 6 t; Qj joins u to xit for every t þ 1 6 j 6 k 1, and Qk joins u to y. We set Wi¼ Qi for every

1 6 i 6 t; Wj¼ hu; Qj; xjt;xjt;Rjt;

v

jg for every t þ 1 6 j 6 k 1, and Wk¼ hu; Qk; y; y; Pktþ11 ;x; Pkt;

v

ki. Then

Suppose that x 2 B. We assume that x ¼

v

k. This case is similar to the above.

Case 3.3.2.

v

12 V11. Suppose that x R B. By induction, there is a spanning ðx; B [ f

v

1gÞ-fan fP1;P2; . . . ;Pktþ1g of G1 F1such that Pijoins x to

v

tþifor every 1 6 i 6 k t and Pktþ1joins x to

v

1. Without loss of generality, we set Pi¼ hx; xi;Ri;

v

tþii for every 1 6 i 6 k t. We set H ¼ A [ fxij1 6 i 6 k tg [ fxg. Let fQ1;Q2; . . . ;Qkg be a spanning ðu; HÞ-fan of G0 F0such that Q1 joins u to x; Qi joins u to

v

i for every 2 6 i 6 t, and Qj joins u to xit for every t þ 1 6 j 6 k. We set W1¼ hu; Q1; x; x; P_ktþ1;

v

1i; Wi¼ Qi for every 2 6 i 6 t, and Wj¼ hu; Qj; xjt;xjt;Rjt;

v

ji for every t þ 1 6 j 6 k. Then fW1;W2; . . . ;Wkg forms a spanning ðu; SÞ-fan of G F.

Suppose that x 2 B. We assume that x ¼

v

k. The case is similar to the above.

Case 4. jT \ V1

0j ¼ k 1. WithCase 2, we consider that jTj P 2.

Case 4.1. jF0j ¼ jFj. We have jF1j ¼ 0 and jF2j ¼ 0. Let x be a node in V10 T. By induction, there exists a spanning ðx; T [ f

ugÞ-fan fP1;P2; . . . ;Pkg of G1such that P1joins x to u, and Pijoins x to

v

ifor every 2 6 i 6 k. We set Pi¼ hx; xi;Ri;

v

ii for every

2 6 i 6 k.

Case 4.1.1.

v

12 V01. We set H ¼ f

v

1g [ fxij2 6 i 6 k 1g. Let fQ1;Q2; . . . ;Qk1g be a spanning ðu; HÞ-fan of G0 F0such that

Q1joins u to

v

1, and Qijoins u to xifor every 2 6 i 6 k 1. We set W1¼ Q1;Wi¼ hu; Qi; xi;xi;Ri;

v

ii for every 2 6 i 6 k 1,

and Wk¼ hu; u; P11 ;x; Pk;

v

Case 4.1.2.

v

12 V11and

v

12 VðP1Þ. We set P1¼ hx; Z1;y;

v

1;Z2; ui. Let H ¼ fyg [ fxij2 6 i 6 k 1g. Thus, there exists a

span-ning ðu; HÞ-fan fQ1;Q2; . . . ;Qk1g in G0 F0such that Q1joins u to y, and Qi joins u to xifor every 2 6 i 6 k 1. We set

W1¼ hu; u; Z12 ;

v

1i; Wi¼ hu; Qi; xi;xi;Ri;

v

ii for every 2 6 i 6 k 1, and Wk¼ hu; Q1; y; y; Z11 ;x; Pk;

v

ki. Then

Case 4.1.3.

v

12 V11 and

v

12 VðPiÞ for some 2 6 i 6 k. Without loss of generality, we assume that

v

12 VðPkÞ. We set

Pk¼ hx; Z1;

v

1;y; Z2;

v

ki. Let H ¼ fyg [ fxij2 6 i 6 k 1g. Thus, there exists a spanning ðu; HÞ-fan fQ1;Q2; . . . ;Qk1g in

G0 F0 such that Q1 joins u to y, and Qi joins u to xi for every 2 6 i 6 k 1. We set

W1¼ hu; u; P11 ;x; Z1;

v

1i; Wi¼ hu; Qi; xi;xi;Ri;

v

ii for every 2 6 i 6 k 1, and Wk¼ hu; Q1; y; y; Z2;

v

ki. Then fW1;W2; . . . ;Wkg

forms a spanning ðu; SÞ-fan of G F.

Case 4.2. jF0j < jFj and jF1j < jFj. Since jV10j ¼ 2 n2_>

ðn 1Þðn 4Þ þ ðn 2Þ P ðn 1ÞjFj þ ðk 2Þ if n P 4, there exists a node x in V1

(8)

Case 4.2.1.

v

12 V01. There exists a node y 2 V 1

1 fx; ug such that ðy; yÞ R F2. Let fP1;P2; . . . ;Pkg be a spanning ðx; T [ fygÞ-fan

of G1 F1such that P1joins x to y and Pijoins x to

v

ifor every 2 6 i 6 k. We set Pi¼ hx; xi;Ri;

v

ii for every 2 6 i 6 k. Since

jfðz; zÞjz 2 NG1ðxÞg \ F2j 6 1, we assume that ðxi; xiÞ R F2for every 2 6 i 6 k 1. We set H ¼ f

v

1; yg [ fxij2 6 i 6 k 1g. By

induction, there is a spanning ðu; HÞ-fan fQ1;Q2; . . . ;Qkg of G0 F0such that Q1joins u to

v

1, Qi joins u to xi for every

2 6 i 6 k 1, and Qk joins u to y. Let W1¼ Q1;Wi¼ hu; Qi; xi;xi;Ri;

v

ii for every 2 6 i 6 k 1, and

Wk¼ hu; Qk; y; y; P11 ;x; Pk;

v

Case 4.2.2.

v

12 V11. Let fP1;P2; . . . ;Pkg be a spanning ðx; SÞ-fan of G1 F1such that Pijoins x to

v

ifor every 1 6 i 6 k. We set

Pi¼ hx; xi;Ri;

v

ii for every 1 6 i 6 k. We choose an index r in f1; 2; . . . ; k 1g such that ðxi; xiÞ R F2 for every

i 2 f1; 2; . . . ; k 1g frg. We set H ¼ fxiji 2 f1; 2; . . . ; k 1g frgg [ fxg. By induction, there is a spanning ðu; HÞ-fan

fQ1;Q2; . . . ;Qkg of G0 F0 such that Qi joins u to xi for every i 2 f1; 2; . . . ; kg frg, and Qr joins u to x. Let

Wi¼ hu; Qi; xi;xi;Ri;

v

ii for every i 2 f1; 2; . . . ; kg frg and Wr¼ hu; Qr; x; x; Pr;

v

ri. Then fW1;W2; . . . ;Wkg forms a spanning

Case 4.3. jF1j ¼ jFj. We have jF0j ¼ 0 and jF2j ¼ 0. Let x1be a node in V11 fu;

v

1g. ByTheorem 3, there is a hamiltonian path P

of G1 F1joining x1to

v

k. We set P ¼ hx1;P1;

v

2;x2;P3;

v

3; . . . ;xk1;Pk1;

v

ki. Note that xi2 V11for every 1 6 i 6 k 1.

Case 4.3.1.

v

12 V01. We set H ¼ f

v

1g [ fxij1 6 i 6 k 1g. By induction, there is a spanning ðu; HÞ-fan fQ1;Q2; . . . ;Qkg of G0

suck that Q1joins u to

v

1and Qijoins u to xi1for every 2 6 i 6 k. We set W1¼ Q1and Wi¼ hu; Qi; xi1;xi1;Pi1;

v

ii for every

2 6 i 6 k. Then fW1;W2; . . . ;Wkg forms a spanning ðu; SÞ-fan of G F.

Case 4.3.2.

v

12 V11. Suppose that

v

1 is a node in VðPtÞ for some 1 6 t 6 k. We write Pt¼ hxt;R1;

v

1;xk;R2;

v

ti. Let

H ¼ fxij1 6 i 6 kg. By induction, there exists a spanning ðu; HÞ-fan fQ1;Q2; . . . ;Qkg of G0where Qijoins u to xi1for every

1 6 i 6 k. For every j 2 f1; 2; . . . ; k 1g ftg, we set Wj¼ hu; Qj; xj;xj;Pj;

v

jþ1i. Let Wt¼ hu; Qt; xt;xt;R1;

v

1i and

Wk¼ hu; Qk; xk;xk;R2;

v

ti. Thus, fW1;W2; . . . ;Wkg forms a spanning ðu; SÞ-fan of G F.

5. Conclusion

Computer network topologies are usually represented by graphs where nodes represent processors and edges represent links between processors[28]. In practice, the processors or links in a network may be failure. Thus the fault-tolerant prop-erty become an important issue on network topologies. Many results have been proposed in literature[29–32,26]. In this paper, we have shown that n-dimensional bipartite hypercube-like graphs are f-edge fault k-laceable for every f 6 n 2 and f þ k 6 n. Future work will try to study the fault-tolerant k_{-connectivity and k}_{-laceability for some super spanning}

con-nected graphs and super spanning laceable graphs, respectively. References

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[13] Y.H. Teng, T.L. Kung, L.H. Hsu, The 3*-connected property of pyramid networks, Computers & Mathematics with Applications 60 (2010) 2360–2363. [14] C.H. Chang, C.K. Lin, H.M. Huang, L.H. Hsu, The super laceability of the hypercube, Information Processing Letters 92 (2004) 15–21.

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[16] S.A. Mane, B.N. Waphare, Regular connected bipancyclic spanning subgraphs of hypercubes, Computers & Mathematics with Applications 62 (2011) 3551–3554.

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[18] S. Abraham, K. Padmanabhan, The twisted cube topology for multiprocessors: a study in network asymmetry, Journal of Parallel and Distributed Computing 13 (1991) 104–110.

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