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Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant
Theoretical Computer Science 589 (2015) 47–52
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Theoretical Computer Science www.elsevier.com/locate/tcs
Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant Che-Nan Kuo Department of Animation and Game Design, TOKO University, No. 51, Sec. 2, University Road, Pu-Tzu City, ChiaYi County 61363, Taiwan
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Article history: Received 19 August 2014 Received in revised form 2 April 2015 Accepted 10 April 2015 Available online 18 April 2015 Communicated by S.-Y. Hsieh Keywords: Interconnection networks Hypercubes Folded hypercubes Embedding Cycle Fault-tolerant Fault-free
a b s t r a c t The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. An n-dimensional folded hypercube (FQ n for short) for any odd n is known to be bipartite. In this paper, let f be a faulty vertex in FQ n . It has been shown that (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. In terms of every edge lies on a fault-free cycle of every odd length in FQ n − { f }, our result improves the result of Cheng et al. (2013) where odd cycle length up to 2n − 3. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Design of interconnection networks (networks for short) is an important integral part of parallel processing and distributed systems. The hypercube is a well-known interconnection network model. The hypercube has several excellent properties, such as recursive structure, regularity, symmetry, small diameter, short mean internode distance, low degree, and much smaller edge complexity, which are very important for designing massively parallel or distributed systems [16]. Numerous variants of the hypercube have been proposed in the literature [3,4,19]. One variant that has been the focus of a great deal of research is the folded hypercube, which can be constructed from a hypercube by adding a link to every pair of nodes that are the farthest apart, i.e., two nodes with complementary addresses. The folded hypercube has been shown to be able to improve the system’s performance over a regular hypercube in many measurements, such as diameter, fault diameter, connectivity, and so on [3,22]. An important feature of an interconnection network is its ability to efficiently simulate algorithms designed for other architectures. Such a simulation can be formulated as network embedding. An embedding of a guest network G into a host network H is defined as a one-to-one mapping f from nodes in G into nodes in H so that a link of G corresponds to a path of H under f [16]. The embedding strategy allows us to emulate the effect of a guest graph on a host graph. Then, algorithms developed for a guest graph can also be executed well on the host graph. Linear arrays and rings, which are two of the most fundamental networks for parallel and distributed computation, are suitable for designing simple algorithms with low communication costs. Numerous efficient algorithms designed on linear arrays and rings for solving various algebraic problems and graph problems can be found in [16]. These algorithms can
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.tcs.2015.04.012 0304-3975/© 2015 Elsevier B.V. All rights reserved.
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be used as control/data flow structures for distributed computing in arbitrary networks. An application of longest paths to a practical problem was encountered in the on-line optimization of a complex Flexible Manufacturing System [1]. These applications motivate the embedding of paths and cycles in networks. Since faults may occur when a network is put into use, it is practically meaningful and important to consider faulty networks. Previously, the problem of fault-tolerant cycle embedding on an n-dimensional folded hypercube FQ n has been studied in [7–9,11–15,22,25,26]. Let F F v and F F e be the sets of faulty vertices and faulty edges of FQ n . Dajin Wang [22] showed that FQ n − F F e 1 contains a Hamiltonian cycle of length 2n if | F F e | ≤ n − 1. Ma [18] showed that FQ n − F F e contains a Hamiltonian cycle of length 2n where each vertex is incident with at least two fault-free edges, when | F F e | ≤ 2n − 3. Hsieh [8] showed that FQ n − F F e remains Hamiltonian-connected if | F F e | ≤ n − 2, where n ≥ 2 is even, and showed that FQ n − F F e remains strongly (respectively, hyper) Hamiltonian-laceable if | F F e | ≤ n − 1 (respectively, | F F e | ≤ n − 2), where n ≥ 3 is odd. Fu [5] showed that FQ n − F F e − F F v contains a cycle of length at least 2n − 2| F F v | if | F F e | ≤ n − 1 and | F F v | + | F F e | ≤ 2n − 4. Xu [24] showed that every edge of FQ n lies on a cycle of every even length from 4 to 2n ; if n is even, every edge of FQ n also lies on a cycle of every odd length from n + 1 to 2n − 1. After that Xu [25] extended the above result to show that every fault-free edge of FQ n − F F e lies on a cycle of every even length from 4 to 2n ; if n is even, every edge of FQ n − F F e also lies on a cycle of every odd length from n + 1 to 2n − 1, where | F F e | ≤ n − 1. Recently, Cheng [2] showed that every fault-free edge of FQ n − F F v lies on a cycle of every even length from 4 to 2n − 2| F F v | if n ≥ 3, and if n ≥ 2 is even, every edge of FQ n − F F v also lies on a cycle of every odd length from n + 1 to 2n − 2| F F v | − 1, where | F F v | ≤ n − 2. In this paper, we extend Cheng’s [2] results to embedding more cycles on FQ n with faulty vertex f . We obtain the following two properties: 1. Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; 2. Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. Throughout this paper, a number of terms—network and graph, node and vertex, edge and link—are used interchangeably. The remainder of this paper is organized as follows: in Section 2, we provide some necessary definitions and notations. We present our main result in Section 3. Some concluding remarks are given in Section 4. 2. Preliminaries A graph G = ( V , E ) is an ordered pair in which V is a finite set and E is a subset of {(u , v ) | (u , v ) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. We also use V (G ) and E (G ) to denote the vertex set and edge set of G, respectively. Two vertices u and v are adjacent if (u , v ) ∈ E. A graph G = ( V 0 ∪ V 1 , E ) is bipartite if V 0 ∩ V 1 = ∅ and E ⊆ {(x, y ) | x ∈ V 0 and y ∈ V 1 }. A path P [ v 0 , v k ] = v 0 , v 1 , . . . , v k is a sequence of distinct vertices in which any two consecutive vertices are adjacent. We call v 0 and v k the end-vertices of the path. In addition, a path may contain a subpath, denoted as v 0 , v 1 , . . . , v i , P [ v i , v j ], v j , v j +1 , . . . , v k , where P [ v i , v j ] = v i , v i +1 , . . . , v j −1 , v j . The length of a path is the number of edges on the path. A path v 0 , v 1 , . . . , v k forms a cycle if v 0 = v k and v 0 , v 1 , . . . , v k−1 are distinct. A vertex is fault-free if it is not faulty. An edge is fault-free if the two end-vertices and the edge between them are not faulty. A path (respectively, cycle) is fault-free if it contains no faulty edges. A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. A Hamiltonian-laceable graph G = ( V 0 ∪ V 1 , E ) is strong [6] if there is a simple path of length | V 0 | + | V 1 | − 2 between any two nodes of the same partite set. A Hamiltonian-laceable graph G = ( V 0 ∪ V 1 , E ) is hyper-Hamiltonian laceable [17] if for any vertex v ∈ V i , i = 0, 1, there is a Hamiltonian path of G − v 2 between any two vertices of V 1−i . For graph-theoretic terminologies and notations not mentioned here, see [23]. An n-dimensional hypercube Q n can be represented as an undirected graph such that V ( Q n ) consists of 2n vertices which are labeled as binary strings of length n from 00 . . . 0 to 11 . . . 1. Each edge e = (u , v ) ∈ E ( Q n ) connects two vertices u and n
n
v if and only if u and v differ in exactly one bit of their labels, i.e., u = bn bn−1 . . . bk . . . b1 and v = bn bn−1 . . . bk . . . b1 , where bk is the one’s complement of bk , i.e., bk = 1 − i iff bk = i for i = 0, 1. We call that e is an edge of dimension k. Clearly, each vertex connects to exactly n other vertices. In addition, there are 2n−1 edges in each dimension and | E ( Q n )| = n · 2n−1 . Fig. 1 shows a 2-dimensional hypercube Q 2 and a 3-dimensional hypercube Q 3 . Let x = xn xn−1 . . . x1 be an n-bit binary string. For 1 ≤ k ≤ n, we use x(k) (respectively, x¯ ) to denote the binary strings yn yn−1 . . . y 1 such that yk = 1 − xk and xi = y i for all i = k (respectively, y i = 1 − xi for all 1 ≤ i ≤ n). The Hamming distance h(x, y ) between two vertices x and y is the number of different bits in the corresponding strings of both vertices. The Hamming weight hw(x) of x is the number of i’s such that xi = 1. Note that Q n is a bipartite graph with two partite sets {x | hw(x) is odd} and {x | hw(x) is even}. Let d Q n (x, y ) be the distance between two vertices x and y in graph Q n . Clearly, d Q n (x, y ) = h(x, y ).
1 2
The graph obtained by deleting F F e from FQ n . The graph obtained by deleting v from G.
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Fig. 1. Q 2 and Q 3 .
Fig. 2. FQ 2 and FQ 3 , in which edges in E s are represented with dashed lines.
An n-dimensional folded hypercube FQ n is a regular n-dimensional hypercube augmented by adding more edges among its vertices. More specifically, FQ n is obtained by adding an edge (also called complementary edge) to every pair of vertices whose address are complementary to each other; i.e., for a vertex whose address is b = b1 b2 . . . bn , it now has one more edge to vertex b¯ = b¯1 b¯2 . . . b¯n , in addition to its original n edges. So FQ n has 2n−1 more edges than a regular Q n . We call these complementary edges skips, to distinguish them from regular edges, and use E s to denote the set of skips. So the complete edge set E (FQ n ) of a folded hypercube can be expressed as E ( Q n ) ∪ E s . In other words, we can formally define the edges of FQ n as that E (FQ n ) = E ( Q n ) ∪ E s = {e = (u , v ) | d H (u , v ) = 1 ∈ E ( Q n ) or d H (u , v ) = n ∈ E s }. Fig. 2 shows a 2-dimensional folded hypercube FQ 2 and a 3-dimensional folded hypercube FQ 3 . It has been shown that FQ n is (n + 1)-regular, (n + 1)-connected, node-transitive, and edge-transitive [25]. For convenience, FQ n can be represented with ∗ . . ∗ ∗ = ∗n , where ∗ ∈ {0, 1} means the “don’t care” symbol. A reg ∗ . n
ular hypercube Q n can be partitioned into two subcubes Q n−1 along some dimension i, where 1 ≤ i ≤ n. We define the subcubes as Q n0−1 = ∗n−i 0∗i −1 and Q n1−1 = ∗n−i 1∗i −1 , in which the values of the ith bits of the vertices are 0 and 1 respec-
tively. Formally, Q n0−1 (respectively, Q n1−1 ) is a subgraph of FQ n induced by {xn . . . xi . . . x1 ∈ V (FQ n ) | xi = 0} (respectively, {xn . . . xi . . . x1 ∈ V (FQ n ) | xi = 1}). Furthermore, we call non-skip edges between Q n0−1 and Q n1−1 crossing edges, denoted by E c ; that is, E c = {(x, y ) ∈ E ( Q n ) | x ∈ V ( Q n0−1 ) and y ∈ V ( Q n1−1 )}. Lemma 1. (See [11].) An i-partition on FQ n = ∗n , where 1 ≤ i ≤ n, partitions FQ n along dimension i into two (n − 1)-dimensional hypercubes ∗n−i 0∗i −1 ( Q n0−1 ) and ∗n−i 1∗i −1 ( Q n1−1 ). Moreover, all edges in E s of FQ n are between Q n0−1 and Q n1−1 . In the remainder of this section, we consider some previously reported results of path (or cycle) embedding in Q n (or FQ n ), as they are useful to our method. For convenience, let F v (respectively, F F v ) and F e (respectively, F F e ) be the sets of faulty vertices and faulty edges in Q n (respectively, FQ n ). Lemma 2. (See [10].) Q n − F e − F v contains a fault-free cycle of every even length from 4 to 2n − 2| F v | if | F e | + | F v | ≤ n − 2, where n ≥ 3. Lemma 3. (See [21].) Let V 0 and V 1 be the partite sets of a fault-free Q n , where n ≥ 2. Let a and b be two distinct nodes of V 0 , and a and b be two distinct nodes of V 1 . Then, there exist two node-disjoint paths P [a, a ] and P [b, b ] spanning V ( Q n ), i.e., V ( P [a, a ]) ∪ V ( P [b, b ]) = V ( Q n ). Lemma 4. (See [20].) Q n − F e remains hyper-Hamiltonian-laceable if | F e | ≤ n − 3, where n ≥ 3.
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Fig. 3. An illustration of Case 1.1 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
Lemma 5. (See [2].) Every edge of FQ n − F F v lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2| F F v | if | F F v | ≤ n − 2, where n ≥ 3. Lemma 6. (See [2].) Every edge of FQ n − F F v lies on a fault-free cycle of every odd length l with 5 ≤ l ≤ 2n − 3 if | F F v | = 1, where n ≥ 4 is even. 3. Every edge lies on cycles embedding in folded hypercubes Let f be a faulty vertex in an n-dimensional folded hypercube FQ n . In this section, we show that (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. Lemma 7. Let f be a faulty vertex in FQ n for n ≥ 3, then every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2. Proof. Because of n ≥ 3 and | F F v | = 1 ≤ n − 2, by Lemma 5, the result is true. Therefore, every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2, where n ≥ 3. 2 Lemma 8. Let f be a faulty vertex in FQ n for n ≥ 2 is even, then every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1. Proof. It is easy to check that the lemma holds for n = 2. We now consider the conditions for n ≥ 4 is even. By Lemma 6, the result that every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 3 can be derived. Therefore, we have to consider the condition for l = 2n − 1. Let e be any fault-free edge in FQ n . By Lemma 1, we can execute an i-partition on FQ n along dimension i, where 1 ≤ i ≤ n, into two (n − 1)-dimensional hypercubes Q n0−1 and Q n1−1 such that e ∈ E c or e ∈ E s . Since the number of faulty vertex is 1, without loss of generality, we may assume that i = n and f ∈ V ( Q n0−1 ). Then, we consider the following cases. Case 1.
e ∈ Ec In this case, let e = (u , u (n) ). By Lemma 2, there exists a fault-free cycle C 0 of length 2n − 2 in Q n0−1 − { f }. Note that the vertex u ∈ V ( Q n0−1 ) is uncertain in C 0 . Then, we consider the following scenarios.
Case 1.1: u ∈ C 0 In this case, let p be the fault-free vertex in Q n0−1 and p ∈ / C 0 .3 Note that p and f have different partite sets in Q n0−1 , and { p , p (n) } are adjacent vertices of p in Q n1−1 . Let v be the adjacent vertex of u in C 0 such that v (n) = p 4 in Q n1−1 . Therefore, cycle C 0 can be represented as u , P [u , v ], v , u . Then, we have the following scenarios.
3 4
Since | V ( Q n0−1 )| − | V (C 0 )| − |{ f }| = 1, there must exist such a vertex p in Q n0−1 − { f }. Since the number of adjacent vertices with u in C 0 is 2, there must exist such a vertex v in Q n0−1 .
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Fig. 4. An illustration of Case 1.2 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
Case 1.1.1: u (n) = p By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [ p (n) , u (n) ] and R 1 [ v (n) , p ] with lengths total are 2n−1 − 2. Therefore, u , P [u , v ], v , v (n) , R 1 [ v (n) , p ], p , p , p (n) , R 0 [ p (n) , u (n) ], u (n) , u
forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 3(a)). Case 1.1.2: u (n) = p By Lemma 4, we can construct a path R [ v (n) , p (n) ] of length 2n−1 − 2 in Q n1−1 − { p }. Therefore,
u , P [u , v ], v , v (n) , R [ v (n) , p (n) ], p (n) , p , p , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 3(b)).
Case 1.2: u ∈ / C0 In Q n0−1 , we can find an edge (x, y ) in C 0 such that {x(n) , y (n) } ∩ {u } = ∅ in Q n1−1 . Note that cycle C 0 can be represented as x, P [x, y ], y , x . By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [u , x(n) ]
and R 1 [ y (n) , u (n) ] with lengths total are 2n−1 − 2. Therefore, u , u , R 0 [u , x(n) ], x(n) , x, P [x, y ], y , y (n) , R 1 [ y (n) , u (n) ], u (n) , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 4).
Case 2.
e ∈ Es In this case, let e = (u , u ). By Lemma 2, there exists a fault-free cycle C 0 of length 2n − 2 in Q n0−1 − { f }. Note that the vertex u ∈ V ( Q n0−1 ) is uncertain in C 0 . Then, we consider the following scenarios.
Case 2.1: u ∈ C 0 In this case, let p be the fault-free vertex in Q n0−1 and p ∈ / C 0 . Note that p and f have different partite
sets in Q n0−1 , and { p , p (n) } are adjacent vertices of p in Q n1−1 . Let v be the adjacent vertex of u in C 0 such that v = p (n) in Q n1−1 . Therefore, cycle C 0 can be represented as u , P [u , v ], v , u . Then, we have the following scenarios. Case 2.1.1: u = p (n) By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [ v , p (n) ] and R 1 [ p , u ] with lengths total
are 2n−1 − 2. Therefore, u , P [u , v ], v , v , R 0 [ v , p (n) ], p (n) , p , p , R 1 [ p , u ], u , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 5(a)).
Case 2.1.2: u = p (n) By Lemma 4, we can construct a path R [ v , p ] of length 2n−1 − 2 in Q n1−1 −{u }. Therefore, u , P [u , v ], v , v , R [ v , p ], p , p , u , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 +(2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 5(b)). Case 2.2: u ∈ / C0 This proof is similar to that in Case 1.2. By combining the above cases, we complete the proof.
2
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Fig. 5. An illustration of Case 2.1 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
By Lemmas 7 and 8, we have the following theorem. Theorem 1. Let f be a faulty vertex in FQ n . Then, every edge of FQ n − { f } lies on a fault-free cycle of every even length from 4 to 2n − 2 where n ≥ 3 and, furthermore, also every odd length from n + 1 to 2n − 1 if n ≥ 2 is even. 4. Concluding remarks Fault-tolerant is an important research topic in the area of the multi-process computer systems, and many studies have focus on the vertex fault-tolerant properties of some specific networks. In this paper, we extend Cheng’s result [2] to embed more cycles on FQ n with faulty vertex f as follows: (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. References [1] N. Ascheuer, Hamiltonian path problems in the on-line optimization of flexible manufacturing systems, Ph.D. Thesis, University of Technology, Berlin, Germany, 1995. [2] D. Cheng, R.S. Hao, Y.Q. Feng, Cycles embedding on folded hypercubes with faulty nodes, Discrete Appl. Math. 161 (18) (2013) 2894–2900. [3] A. EI-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. Parallel Distrib. Syst. 2 (1) (1991) 31–42. [4] A.H. Esfahanian, L.M. Ni, B.E. Sagan, The twisted n-cube with application to multiprocessing, IEEE Trans. Comput. 40 (1991) 88. [5] J.S. Fu, Fault-free cycles in folded hypercubes with more faulty elements, Inform. Process. Lett. 108 (5) (2008) 261–263. [6] S.Y. Hsieh, G.H. Chen, C.W. Ho, Hamiltonian-laceability of star graphs, Networks 36 (4) (2000) 225–232. [7] S.Y. Hsieh, Fault-tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges, Parallel Comput. 32 (1) (2006) 84–91. [8] S.Y. Hsieh, Some edge-fault-tolerant properties of the folded hypercube, Networks 51 (2) (2008) 92–101. [9] S.Y. Hsieh, A note on cycle embedding in folded hypercubes with faulty elements, Inform. Process. Lett. 108 (2) (2008) 81. [10] S.Y. Hsieh, T.H. Shen, Edge-bipancyclicity of a hypercube with faulty vertices and edges, Discrete Appl. Math. 156 (2) (2008) 1802–1808. [11] S.Y. Hsieh, C.N. Kuo, Hamilton-connectivity and strongly Hamiltonian-laceability of folded hypercubes, Comput. Math. Appl. 53 (7) (2007) 1040–1044. [12] S.Y. Hsieh, C.N. Kuo, H.H. Chou, A further result on fault-free cycles in faulty folded hypercubes, Inform. Process. Lett. 110 (2) (2009) 41–43. [13] S.Y. Hsieh, C.N. Kuo, H.L. Huang, 1-vertex-fault-tolerant cycles embedding on folded hypercubes, Discrete Appl. Math. 157 (14) (2009) 3094–3098. [14] C.N. Kuo, S.Y. Hsieh, Pancyclicity and bipancyclicity of conditional faulty folded hypercubes, Inform. Sci. 180 (15) (2010) 2904–2914. [15] C.N. Kuo, H.H. Chou, N.W. Chang, S.Y. Hsieh, Fault-tolerant path embedding in folded hypercubes with both node and edge faults, Theoret. Comput. Sci. 574 (2013) 82–91. [16] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays · Trees · Hypercubes, Morgan Kaufman, San Mateo, CA, USA, 1992. [17] M. Lewinter, W. Widulski, Hyper-Hamilton laceable and caterpillar-spannable product graphs, Comput. Math. Appl. 34 (11) (1997) 99–104. [18] M.J. Ma, J.M. Xu, Z.Z. Du, Edge-fault-tolerant hamiltonicity of folded hypercubes, J. Univ. Sci. Technol. China 36 (3) (2007) 244–248. [19] F.P. Preparata, J. Vuillemin, The cube-connected cycles: a versatile network for parallel computation, Commun. ACM 24 (1981) 300–309. [20] C.H. Tsai, Jimmy J.M. Tan, T. Liang, L.H. Hsu, Fault-tolerant Hamiltonian laceability of hypercubes, Inform. Process. Lett. 83 (2002) 301–306. [21] C.H. Tsai, Linear array and ring embedding in conditional faulty hypercubes, Theoret. Comput. Sci. 314 (2004) 431–443. [22] D. Wang, Embedding Hamiltonian cycles into folded hypercubes with faulty links, J. Parallel Distrib. Comput. 61 (2001) 545–564. [23] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, USA, 2001. [24] J.M. Xu, M. Ma, Cycles in folded hypercubes, Appl. Math. Lett. 19 (2006) 140–145. [25] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant properties of hypercubes and folded hypercubes, Australas. J. Combin. 35 (2006) 7–16. [26] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant Hamiltonicity of folded hypercubes, J. Univ. Sci. Technol. China 36 (3) (2007) 244–248.
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Theoretical Computer Science www.elsevier.com/locate/tcs
Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant Che-Nan Kuo Department of Animation and Game Design, TOKO University, No. 51, Sec. 2, University Road, Pu-Tzu City, ChiaYi County 61363, Taiwan
a r t i c l e
i n f o
Article history: Received 19 August 2014 Received in revised form 2 April 2015 Accepted 10 April 2015 Available online 18 April 2015 Communicated by S.-Y. Hsieh Keywords: Interconnection networks Hypercubes Folded hypercubes Embedding Cycle Fault-tolerant Fault-free
a b s t r a c t The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. An n-dimensional folded hypercube (FQ n for short) for any odd n is known to be bipartite. In this paper, let f be a faulty vertex in FQ n . It has been shown that (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. In terms of every edge lies on a fault-free cycle of every odd length in FQ n − { f }, our result improves the result of Cheng et al. (2013) where odd cycle length up to 2n − 3. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Design of interconnection networks (networks for short) is an important integral part of parallel processing and distributed systems. The hypercube is a well-known interconnection network model. The hypercube has several excellent properties, such as recursive structure, regularity, symmetry, small diameter, short mean internode distance, low degree, and much smaller edge complexity, which are very important for designing massively parallel or distributed systems [16]. Numerous variants of the hypercube have been proposed in the literature [3,4,19]. One variant that has been the focus of a great deal of research is the folded hypercube, which can be constructed from a hypercube by adding a link to every pair of nodes that are the farthest apart, i.e., two nodes with complementary addresses. The folded hypercube has been shown to be able to improve the system’s performance over a regular hypercube in many measurements, such as diameter, fault diameter, connectivity, and so on [3,22]. An important feature of an interconnection network is its ability to efficiently simulate algorithms designed for other architectures. Such a simulation can be formulated as network embedding. An embedding of a guest network G into a host network H is defined as a one-to-one mapping f from nodes in G into nodes in H so that a link of G corresponds to a path of H under f [16]. The embedding strategy allows us to emulate the effect of a guest graph on a host graph. Then, algorithms developed for a guest graph can also be executed well on the host graph. Linear arrays and rings, which are two of the most fundamental networks for parallel and distributed computation, are suitable for designing simple algorithms with low communication costs. Numerous efficient algorithms designed on linear arrays and rings for solving various algebraic problems and graph problems can be found in [16]. These algorithms can
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.tcs.2015.04.012 0304-3975/© 2015 Elsevier B.V. All rights reserved.
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be used as control/data flow structures for distributed computing in arbitrary networks. An application of longest paths to a practical problem was encountered in the on-line optimization of a complex Flexible Manufacturing System [1]. These applications motivate the embedding of paths and cycles in networks. Since faults may occur when a network is put into use, it is practically meaningful and important to consider faulty networks. Previously, the problem of fault-tolerant cycle embedding on an n-dimensional folded hypercube FQ n has been studied in [7–9,11–15,22,25,26]. Let F F v and F F e be the sets of faulty vertices and faulty edges of FQ n . Dajin Wang [22] showed that FQ n − F F e 1 contains a Hamiltonian cycle of length 2n if | F F e | ≤ n − 1. Ma [18] showed that FQ n − F F e contains a Hamiltonian cycle of length 2n where each vertex is incident with at least two fault-free edges, when | F F e | ≤ 2n − 3. Hsieh [8] showed that FQ n − F F e remains Hamiltonian-connected if | F F e | ≤ n − 2, where n ≥ 2 is even, and showed that FQ n − F F e remains strongly (respectively, hyper) Hamiltonian-laceable if | F F e | ≤ n − 1 (respectively, | F F e | ≤ n − 2), where n ≥ 3 is odd. Fu [5] showed that FQ n − F F e − F F v contains a cycle of length at least 2n − 2| F F v | if | F F e | ≤ n − 1 and | F F v | + | F F e | ≤ 2n − 4. Xu [24] showed that every edge of FQ n lies on a cycle of every even length from 4 to 2n ; if n is even, every edge of FQ n also lies on a cycle of every odd length from n + 1 to 2n − 1. After that Xu [25] extended the above result to show that every fault-free edge of FQ n − F F e lies on a cycle of every even length from 4 to 2n ; if n is even, every edge of FQ n − F F e also lies on a cycle of every odd length from n + 1 to 2n − 1, where | F F e | ≤ n − 1. Recently, Cheng [2] showed that every fault-free edge of FQ n − F F v lies on a cycle of every even length from 4 to 2n − 2| F F v | if n ≥ 3, and if n ≥ 2 is even, every edge of FQ n − F F v also lies on a cycle of every odd length from n + 1 to 2n − 2| F F v | − 1, where | F F v | ≤ n − 2. In this paper, we extend Cheng’s [2] results to embedding more cycles on FQ n with faulty vertex f . We obtain the following two properties: 1. Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; 2. Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. Throughout this paper, a number of terms—network and graph, node and vertex, edge and link—are used interchangeably. The remainder of this paper is organized as follows: in Section 2, we provide some necessary definitions and notations. We present our main result in Section 3. Some concluding remarks are given in Section 4. 2. Preliminaries A graph G = ( V , E ) is an ordered pair in which V is a finite set and E is a subset of {(u , v ) | (u , v ) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. We also use V (G ) and E (G ) to denote the vertex set and edge set of G, respectively. Two vertices u and v are adjacent if (u , v ) ∈ E. A graph G = ( V 0 ∪ V 1 , E ) is bipartite if V 0 ∩ V 1 = ∅ and E ⊆ {(x, y ) | x ∈ V 0 and y ∈ V 1 }. A path P [ v 0 , v k ] = v 0 , v 1 , . . . , v k is a sequence of distinct vertices in which any two consecutive vertices are adjacent. We call v 0 and v k the end-vertices of the path. In addition, a path may contain a subpath, denoted as v 0 , v 1 , . . . , v i , P [ v i , v j ], v j , v j +1 , . . . , v k , where P [ v i , v j ] = v i , v i +1 , . . . , v j −1 , v j . The length of a path is the number of edges on the path. A path v 0 , v 1 , . . . , v k forms a cycle if v 0 = v k and v 0 , v 1 , . . . , v k−1 are distinct. A vertex is fault-free if it is not faulty. An edge is fault-free if the two end-vertices and the edge between them are not faulty. A path (respectively, cycle) is fault-free if it contains no faulty edges. A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. A Hamiltonian-laceable graph G = ( V 0 ∪ V 1 , E ) is strong [6] if there is a simple path of length | V 0 | + | V 1 | − 2 between any two nodes of the same partite set. A Hamiltonian-laceable graph G = ( V 0 ∪ V 1 , E ) is hyper-Hamiltonian laceable [17] if for any vertex v ∈ V i , i = 0, 1, there is a Hamiltonian path of G − v 2 between any two vertices of V 1−i . For graph-theoretic terminologies and notations not mentioned here, see [23]. An n-dimensional hypercube Q n can be represented as an undirected graph such that V ( Q n ) consists of 2n vertices which are labeled as binary strings of length n from 00 . . . 0 to 11 . . . 1. Each edge e = (u , v ) ∈ E ( Q n ) connects two vertices u and n
n
v if and only if u and v differ in exactly one bit of their labels, i.e., u = bn bn−1 . . . bk . . . b1 and v = bn bn−1 . . . bk . . . b1 , where bk is the one’s complement of bk , i.e., bk = 1 − i iff bk = i for i = 0, 1. We call that e is an edge of dimension k. Clearly, each vertex connects to exactly n other vertices. In addition, there are 2n−1 edges in each dimension and | E ( Q n )| = n · 2n−1 . Fig. 1 shows a 2-dimensional hypercube Q 2 and a 3-dimensional hypercube Q 3 . Let x = xn xn−1 . . . x1 be an n-bit binary string. For 1 ≤ k ≤ n, we use x(k) (respectively, x¯ ) to denote the binary strings yn yn−1 . . . y 1 such that yk = 1 − xk and xi = y i for all i = k (respectively, y i = 1 − xi for all 1 ≤ i ≤ n). The Hamming distance h(x, y ) between two vertices x and y is the number of different bits in the corresponding strings of both vertices. The Hamming weight hw(x) of x is the number of i’s such that xi = 1. Note that Q n is a bipartite graph with two partite sets {x | hw(x) is odd} and {x | hw(x) is even}. Let d Q n (x, y ) be the distance between two vertices x and y in graph Q n . Clearly, d Q n (x, y ) = h(x, y ).
1 2
The graph obtained by deleting F F e from FQ n . The graph obtained by deleting v from G.
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Fig. 1. Q 2 and Q 3 .
Fig. 2. FQ 2 and FQ 3 , in which edges in E s are represented with dashed lines.
An n-dimensional folded hypercube FQ n is a regular n-dimensional hypercube augmented by adding more edges among its vertices. More specifically, FQ n is obtained by adding an edge (also called complementary edge) to every pair of vertices whose address are complementary to each other; i.e., for a vertex whose address is b = b1 b2 . . . bn , it now has one more edge to vertex b¯ = b¯1 b¯2 . . . b¯n , in addition to its original n edges. So FQ n has 2n−1 more edges than a regular Q n . We call these complementary edges skips, to distinguish them from regular edges, and use E s to denote the set of skips. So the complete edge set E (FQ n ) of a folded hypercube can be expressed as E ( Q n ) ∪ E s . In other words, we can formally define the edges of FQ n as that E (FQ n ) = E ( Q n ) ∪ E s = {e = (u , v ) | d H (u , v ) = 1 ∈ E ( Q n ) or d H (u , v ) = n ∈ E s }. Fig. 2 shows a 2-dimensional folded hypercube FQ 2 and a 3-dimensional folded hypercube FQ 3 . It has been shown that FQ n is (n + 1)-regular, (n + 1)-connected, node-transitive, and edge-transitive [25]. For convenience, FQ n can be represented with ∗ . . ∗ ∗ = ∗n , where ∗ ∈ {0, 1} means the “don’t care” symbol. A reg ∗ . n
ular hypercube Q n can be partitioned into two subcubes Q n−1 along some dimension i, where 1 ≤ i ≤ n. We define the subcubes as Q n0−1 = ∗n−i 0∗i −1 and Q n1−1 = ∗n−i 1∗i −1 , in which the values of the ith bits of the vertices are 0 and 1 respec-
tively. Formally, Q n0−1 (respectively, Q n1−1 ) is a subgraph of FQ n induced by {xn . . . xi . . . x1 ∈ V (FQ n ) | xi = 0} (respectively, {xn . . . xi . . . x1 ∈ V (FQ n ) | xi = 1}). Furthermore, we call non-skip edges between Q n0−1 and Q n1−1 crossing edges, denoted by E c ; that is, E c = {(x, y ) ∈ E ( Q n ) | x ∈ V ( Q n0−1 ) and y ∈ V ( Q n1−1 )}. Lemma 1. (See [11].) An i-partition on FQ n = ∗n , where 1 ≤ i ≤ n, partitions FQ n along dimension i into two (n − 1)-dimensional hypercubes ∗n−i 0∗i −1 ( Q n0−1 ) and ∗n−i 1∗i −1 ( Q n1−1 ). Moreover, all edges in E s of FQ n are between Q n0−1 and Q n1−1 . In the remainder of this section, we consider some previously reported results of path (or cycle) embedding in Q n (or FQ n ), as they are useful to our method. For convenience, let F v (respectively, F F v ) and F e (respectively, F F e ) be the sets of faulty vertices and faulty edges in Q n (respectively, FQ n ). Lemma 2. (See [10].) Q n − F e − F v contains a fault-free cycle of every even length from 4 to 2n − 2| F v | if | F e | + | F v | ≤ n − 2, where n ≥ 3. Lemma 3. (See [21].) Let V 0 and V 1 be the partite sets of a fault-free Q n , where n ≥ 2. Let a and b be two distinct nodes of V 0 , and a and b be two distinct nodes of V 1 . Then, there exist two node-disjoint paths P [a, a ] and P [b, b ] spanning V ( Q n ), i.e., V ( P [a, a ]) ∪ V ( P [b, b ]) = V ( Q n ). Lemma 4. (See [20].) Q n − F e remains hyper-Hamiltonian-laceable if | F e | ≤ n − 3, where n ≥ 3.
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Fig. 3. An illustration of Case 1.1 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
Lemma 5. (See [2].) Every edge of FQ n − F F v lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2| F F v | if | F F v | ≤ n − 2, where n ≥ 3. Lemma 6. (See [2].) Every edge of FQ n − F F v lies on a fault-free cycle of every odd length l with 5 ≤ l ≤ 2n − 3 if | F F v | = 1, where n ≥ 4 is even. 3. Every edge lies on cycles embedding in folded hypercubes Let f be a faulty vertex in an n-dimensional folded hypercube FQ n . In this section, we show that (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. Lemma 7. Let f be a faulty vertex in FQ n for n ≥ 3, then every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2. Proof. Because of n ≥ 3 and | F F v | = 1 ≤ n − 2, by Lemma 5, the result is true. Therefore, every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2, where n ≥ 3. 2 Lemma 8. Let f be a faulty vertex in FQ n for n ≥ 2 is even, then every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1. Proof. It is easy to check that the lemma holds for n = 2. We now consider the conditions for n ≥ 4 is even. By Lemma 6, the result that every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 3 can be derived. Therefore, we have to consider the condition for l = 2n − 1. Let e be any fault-free edge in FQ n . By Lemma 1, we can execute an i-partition on FQ n along dimension i, where 1 ≤ i ≤ n, into two (n − 1)-dimensional hypercubes Q n0−1 and Q n1−1 such that e ∈ E c or e ∈ E s . Since the number of faulty vertex is 1, without loss of generality, we may assume that i = n and f ∈ V ( Q n0−1 ). Then, we consider the following cases. Case 1.
e ∈ Ec In this case, let e = (u , u (n) ). By Lemma 2, there exists a fault-free cycle C 0 of length 2n − 2 in Q n0−1 − { f }. Note that the vertex u ∈ V ( Q n0−1 ) is uncertain in C 0 . Then, we consider the following scenarios.
Case 1.1: u ∈ C 0 In this case, let p be the fault-free vertex in Q n0−1 and p ∈ / C 0 .3 Note that p and f have different partite sets in Q n0−1 , and { p , p (n) } are adjacent vertices of p in Q n1−1 . Let v be the adjacent vertex of u in C 0 such that v (n) = p 4 in Q n1−1 . Therefore, cycle C 0 can be represented as u , P [u , v ], v , u . Then, we have the following scenarios.
3 4
Since | V ( Q n0−1 )| − | V (C 0 )| − |{ f }| = 1, there must exist such a vertex p in Q n0−1 − { f }. Since the number of adjacent vertices with u in C 0 is 2, there must exist such a vertex v in Q n0−1 .
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Fig. 4. An illustration of Case 1.2 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
Case 1.1.1: u (n) = p By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [ p (n) , u (n) ] and R 1 [ v (n) , p ] with lengths total are 2n−1 − 2. Therefore, u , P [u , v ], v , v (n) , R 1 [ v (n) , p ], p , p , p (n) , R 0 [ p (n) , u (n) ], u (n) , u
forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 3(a)). Case 1.1.2: u (n) = p By Lemma 4, we can construct a path R [ v (n) , p (n) ] of length 2n−1 − 2 in Q n1−1 − { p }. Therefore,
u , P [u , v ], v , v (n) , R [ v (n) , p (n) ], p (n) , p , p , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 3(b)).
Case 1.2: u ∈ / C0 In Q n0−1 , we can find an edge (x, y ) in C 0 such that {x(n) , y (n) } ∩ {u } = ∅ in Q n1−1 . Note that cycle C 0 can be represented as x, P [x, y ], y , x . By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [u , x(n) ]
and R 1 [ y (n) , u (n) ] with lengths total are 2n−1 − 2. Therefore, u , u , R 0 [u , x(n) ], x(n) , x, P [x, y ], y , y (n) , R 1 [ y (n) , u (n) ], u (n) , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 4).
Case 2.
e ∈ Es In this case, let e = (u , u ). By Lemma 2, there exists a fault-free cycle C 0 of length 2n − 2 in Q n0−1 − { f }. Note that the vertex u ∈ V ( Q n0−1 ) is uncertain in C 0 . Then, we consider the following scenarios.
Case 2.1: u ∈ C 0 In this case, let p be the fault-free vertex in Q n0−1 and p ∈ / C 0 . Note that p and f have different partite
sets in Q n0−1 , and { p , p (n) } are adjacent vertices of p in Q n1−1 . Let v be the adjacent vertex of u in C 0 such that v = p (n) in Q n1−1 . Therefore, cycle C 0 can be represented as u , P [u , v ], v , u . Then, we have the following scenarios. Case 2.1.1: u = p (n) By Lemma 3, Q n1−1 contains two node-disjoint paths R 0 [ v , p (n) ] and R 1 [ p , u ] with lengths total
are 2n−1 − 2. Therefore, u , P [u , v ], v , v , R 0 [ v , p (n) ], p (n) , p , p , R 1 [ p , u ], u , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 + (2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 5(a)).
Case 2.1.2: u = p (n) By Lemma 4, we can construct a path R [ v , p ] of length 2n−1 − 2 in Q n1−1 −{u }. Therefore, u , P [u , v ], v , v , R [ v , p ], p , p , u , u forms a cycle of length l with l = (2n−1 − 2) − 1 + 4 +(2n−1 − 2) = 2n − 1 which contains the edge e in FQ n − { f } (see Fig. 5(b)). Case 2.2: u ∈ / C0 This proof is similar to that in Case 1.2. By combining the above cases, we complete the proof.
2
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Fig. 5. An illustration of Case 2.1 in the proof of Lemma 8, in which the vertices with indefinite partite sets are filled with gray colors.
By Lemmas 7 and 8, we have the following theorem. Theorem 1. Let f be a faulty vertex in FQ n . Then, every edge of FQ n − { f } lies on a fault-free cycle of every even length from 4 to 2n − 2 where n ≥ 3 and, furthermore, also every odd length from n + 1 to 2n − 1 if n ≥ 2 is even. 4. Concluding remarks Fault-tolerant is an important research topic in the area of the multi-process computer systems, and many studies have focus on the vertex fault-tolerant properties of some specific networks. In this paper, we extend Cheng’s result [2] to embed more cycles on FQ n with faulty vertex f as follows: (1) Every edge of FQ n − { f } lies on a fault-free cycle of every even length l with 4 ≤ l ≤ 2n − 2 where n ≥ 3; (2) Every edge of FQ n − { f } lies on a fault-free cycle of every odd length l with n + 1 ≤ l ≤ 2n − 1, where n ≥ 2 is even. References [1] N. Ascheuer, Hamiltonian path problems in the on-line optimization of flexible manufacturing systems, Ph.D. Thesis, University of Technology, Berlin, Germany, 1995. [2] D. Cheng, R.S. Hao, Y.Q. Feng, Cycles embedding on folded hypercubes with faulty nodes, Discrete Appl. Math. 161 (18) (2013) 2894–2900. [3] A. EI-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. Parallel Distrib. Syst. 2 (1) (1991) 31–42. [4] A.H. Esfahanian, L.M. Ni, B.E. Sagan, The twisted n-cube with application to multiprocessing, IEEE Trans. Comput. 40 (1991) 88. [5] J.S. Fu, Fault-free cycles in folded hypercubes with more faulty elements, Inform. Process. Lett. 108 (5) (2008) 261–263. [6] S.Y. Hsieh, G.H. Chen, C.W. Ho, Hamiltonian-laceability of star graphs, Networks 36 (4) (2000) 225–232. [7] S.Y. Hsieh, Fault-tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges, Parallel Comput. 32 (1) (2006) 84–91. [8] S.Y. Hsieh, Some edge-fault-tolerant properties of the folded hypercube, Networks 51 (2) (2008) 92–101. [9] S.Y. Hsieh, A note on cycle embedding in folded hypercubes with faulty elements, Inform. Process. Lett. 108 (2) (2008) 81. [10] S.Y. Hsieh, T.H. Shen, Edge-bipancyclicity of a hypercube with faulty vertices and edges, Discrete Appl. Math. 156 (2) (2008) 1802–1808. [11] S.Y. Hsieh, C.N. Kuo, Hamilton-connectivity and strongly Hamiltonian-laceability of folded hypercubes, Comput. Math. Appl. 53 (7) (2007) 1040–1044. [12] S.Y. Hsieh, C.N. Kuo, H.H. Chou, A further result on fault-free cycles in faulty folded hypercubes, Inform. Process. Lett. 110 (2) (2009) 41–43. [13] S.Y. Hsieh, C.N. Kuo, H.L. Huang, 1-vertex-fault-tolerant cycles embedding on folded hypercubes, Discrete Appl. Math. 157 (14) (2009) 3094–3098. [14] C.N. Kuo, S.Y. Hsieh, Pancyclicity and bipancyclicity of conditional faulty folded hypercubes, Inform. Sci. 180 (15) (2010) 2904–2914. [15] C.N. Kuo, H.H. Chou, N.W. Chang, S.Y. Hsieh, Fault-tolerant path embedding in folded hypercubes with both node and edge faults, Theoret. Comput. Sci. 574 (2013) 82–91. [16] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays · Trees · Hypercubes, Morgan Kaufman, San Mateo, CA, USA, 1992. [17] M. Lewinter, W. Widulski, Hyper-Hamilton laceable and caterpillar-spannable product graphs, Comput. Math. Appl. 34 (11) (1997) 99–104. [18] M.J. Ma, J.M. Xu, Z.Z. Du, Edge-fault-tolerant hamiltonicity of folded hypercubes, J. Univ. Sci. Technol. China 36 (3) (2007) 244–248. [19] F.P. Preparata, J. Vuillemin, The cube-connected cycles: a versatile network for parallel computation, Commun. ACM 24 (1981) 300–309. [20] C.H. Tsai, Jimmy J.M. Tan, T. Liang, L.H. Hsu, Fault-tolerant Hamiltonian laceability of hypercubes, Inform. Process. Lett. 83 (2002) 301–306. [21] C.H. Tsai, Linear array and ring embedding in conditional faulty hypercubes, Theoret. Comput. Sci. 314 (2004) 431–443. [22] D. Wang, Embedding Hamiltonian cycles into folded hypercubes with faulty links, J. Parallel Distrib. Comput. 61 (2001) 545–564. [23] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, USA, 2001. [24] J.M. Xu, M. Ma, Cycles in folded hypercubes, Appl. Math. Lett. 19 (2006) 140–145. [25] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant properties of hypercubes and folded hypercubes, Australas. J. Combin. 35 (2006) 7–16. [26] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant Hamiltonicity of folded hypercubes, J. Univ. Sci. Technol. China 36 (3) (2007) 244–248.