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A further result on fault-free cycles in faulty folded hypercubes
Information Processing Letters 110 (2009) 41–43
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
A further result on fault-free cycles in faulty folded hypercubes Sun-Yuan Hsieh a , Che-Nan Kuo a , Hsin-Hung Chou b,∗ a b
Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan Department of Information Management, Chang Jung Christian University, Tainan County, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 2 April 2009 Received in revised form 29 September 2009 Accepted 2 October 2009 Available online 8 October 2009 Communicated by A.A. Bertossi Keywords: Folded hypercubes Interconnection networks Cycle embedding Faulty elements
Folded hypercube is a well-known variation of the hypercube structure and can be constructed from a hypercube by adding a link to every pair of nodes with complementary addresses. Let F F v (respectively, F F e ) be the set of faulty nodes (respectively, faulty links) in an n-dimensional folded hypercube F Q n . Fu has showed that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In this paper, we further consider the constraints | F F v | + | F F e | 2n − 4 and | F F e | n that were not covered by Fu’s result. We obtain the same lower bound of the longest fault-free cycle length, 2n − 2| F F v |, under the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Design of interconnection networks (networks for short) is essential to developing the parallel or distributed systems. There various topologies of interconnection networks have been proposed. The interested readers may refer to [2,8,13] for extensive references. Among the proposed network topologies, the hypercube [3] 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 [9]. Some variants of the hypercube have been proposed in the literature [1,5,11]. One of the most popular variants is the folded hypercube, which is an extension of the hypercube and can be constructed by adding a link to every pair of nodes with complementary addresses. The folded hypercube has been shown to be able to im-
*
Corresponding author. E-mail addresses: [email protected] (S.-Y. Hsieh), [email protected] (C.-N. Kuo), [email protected] (H.-H. Chou). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.10.003
© 2009 Elsevier B.V.
All rights reserved.
prove the system’s performance over a regular hypercube in many measurements [1,12]. 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 [9]. Since faults may happen on both nodes and links in a network, it is practically meaningful and important to consider faulty networks. In this paper, we consider faulty folded hypercubes, i.e., folded hypercubes with faulty elements which include faulty nodes and faulty links. A node is fault-free if it is not faulty. A link is fault-free if the communication link between end-nodes is not faulty. A path (cycle) is fault-free if it contains neither faulty nodes nor faulty links. Previously, the problem of fault-tolerant embedding on an n-dimensional folded hypercube F Q n has been studied in [6,7,10,12,14,15]. Let F F v and F F e be the sets of faulty nodes and faulty links of F Q n , respectively. Fu [6] has showed that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In this paper, we further consider the constraints | F F v | + | F F e | 2n − 4
42
S.-Y. Hsieh et al. / Information Processing Letters 110 (2009) 41–43
and | F F e | n that were not covered by Fu’s result. We obtain the same lower bound of the longest fault-free cycle length, 2n − 2| F F v |, under the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. 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. Concluding remarks are given in Section 4. 2. Preliminaries In this paper, a network topology is represented by a simple undirected graph, which is loopless and without multiple edges. We denote the vertex set and the edge set of a graph G by V (G ) and E (G ), respectively. We use an unordered pair of vertices (u , v ) to denote an edge with end-vertices (or end-nodes) u and v. The end-vertices of an edge are said to be incident with the edge, and vice versa. Two vertices which are incident with a common edge are adjacent. Two adjacent vertices are called neighbors each other. Throughout this paper, the terms – network and graph, node and vertex, link and edge – are used interchangeable. A path, denoted by v 0 , v 1 , . . . , v k , is a sequence of distinct vertices v 0 , v 1 , . . . , v k in which any two consecutive vertices are adjacent. We call v 0 and v k the end-vertices (or end-nodes) of the path. 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 . An n-dimensional hypercube Q n is an undirected graph with 2n nodes which are labelled by n-bit binary strings . . . 0 to 11 . . . 1. Two nodes are joined by a link from 00 n
n
if and only if their labels differ by one bit. In particular, a link is called a dimension k link if it links two nodes that differ in the kth bit. An n-dimensional folded hypercube F Q n can be constructed from an n-dimensional hypercube by adding a link to every pair of nodes with complementary addresses, e.g., node x = b1 b2 . . . bn and node x¯ = b¯1 b¯2 . . . b¯n where b¯i represents 1 − b i , the 1’s complement of b i . Thus, F Q n has 2n−1 more links than a regular Q n . We call these extra links skips to distinguish them from regular links. Fig. 1 illustrates a 2-dimensional and a 3-dimensional folded hypercubes. In the following, we introduce some previous results that will be employed later. In an n-dimensional folded hypercube F Q n , let E i be the set of links on dimension i, for 1 i n, and let E s be the set of skips. Xu et al. [15] have shown the result that the folded hypercubes are edgetransitive, Lemma 1, as follows. Lemma 1. (See [15].) There is an automorphism σ of F Q n such that σ ( E i ) = E j for i , j ∈ {1, 2, . . . , n} ∪ {s}. It directly derives the following corollary. Corollary 1. F Q n − E i is isomorphic to Q n for i ∈ {1, 2, . . . , n} ∪ { s }.
Fig. 1. Graphs of F Q 2 and F Q 3 , in which complementary links are drawn by dashed lines.
In an n-dimensional faulty folded hypercube F Q n , let F F v and F F e be the sets of faulty nodes and faulty links of F Q n , respectively. On the problem of finding the lower bound of longest fault-free cycle in F Q n , Fu [6] has shown the result as Lemma 2. Lemma 2. (See [6].) F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In an n-dimensional faulty hypercube F e be the sets of faulty nodes and faulty spectively. On the problem of finding the longest fault-free cycle in Q n , Du et al. [4] result as Lemma 3.
Q n , let F v and links of Q n , relower bound of have shown the
Lemma 3. (See [4].) Q n − F v − F e for n 3 contains a fault-free cycle of length at least 2n − 2| F v | if (1) | F v | + | F e | 2n − 4 and | F e | 2n − 5 and (2) every node in Q n is incident to at least two fault-free links. 3. Fault-free cycle in the faulty folded hypercubes In this section, we present our main result on considering the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links, as shown in Theorem 1. In an n-dimensional faulty folded hypercube F Q n , we call a node k-free if it is incident to at most k fault-free links. Lemma 4. For n 3, if | F F v | + | F F e | 2n − 4, there is at most one 2-free node contained in F Q n .
S.-Y. Hsieh et al. / Information Processing Letters 110 (2009) 41–43
43
Case 2. There is a unique 2-free node u in F Q n and every node in F Q n − {u } is k-free for some k 3. In this case, the number of faulty links incident to u is n − 1. We can choose a link dimension j ∈ {1, 2, . . . , n} ∪ {s} on which there exists at least one faulty link incident to u, such that F Q n − E j is isomorphic to a hypercube Q n . With the same arguments as Case 1, we have that F Q n − E j also satisfies the constraints in Lemma 3. It derives that F Q n − F F v − F F e contains a fault-free cycle of length at least 2n − 2| F F v |. By combining the above two cases, we complete the proof. 2 4. Concluding remarks
Fig. 2. Illustration the proof of Lemma 4.
Proof. Assume that F Q n contains two 2-free nodes u and v. Let us consider a special case that u and v are adjacent. (See Fig. 2.) The least total number of faulty elements is (n − 1) + (n − 1) − 1 = 2n − 3, which contradicts to the assumption that | F F v | + | F F e | 2n − 4. Thus, F Q n contains at most one 2-free node. 2 Theorem 1. F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if (1) | F F v |+| F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. Proof. By Lemma 2, F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | for | F F v | + | F F e | 2n − 4 and | F F e | n − 1. Thus, we only need to consider the constraints | F F v | + | F F e | 2n − 4 and | F F e | n. Since | F F v | + | F F e | 2n − 4 and | F F e | n, we have that | F F v | n − 4. By Lemma 4, we have that F Q n contains at most one 2-free nodes when | F F v | + | F F e | 2n − 4. Then, we consider the following two cases according to the number of 2-free nodes: Case 1. F Q n contains no 2-free node. That is, every node in F Q n is k-free for some k 3. In this case, we can choose a link dimension j ∈ {1, 2, . . . , n} ∪ {s} on which there exists at least one faulty link. By Corollary 1, we have that F Q n − E j is isomorphic to a hypercube Q n . Let F v (respectively, F e ) be the set of faulty nodes (respectively, faulty links) in Q n . Then, | F v | = | F F v |, | F e | = | F F e − E j | 2n − 5 and | F v | + | F e | = | F F v | + | F e | 2n − 4. Since every node in F Q n is k-free for some k 3, every node in F Q n − E j is incident to at least two fault-free links. By Lemma 3, there exists a fault-free cycle of length at least 2n − 2| F v |(= 2n − 2| F F v |) in Q n − F v − F e . Since Q n − F v − F e ⊂ F Q n − F F v − F F e , we obtain that F Q n − F F v − F F e contains a fault-free cycle of length at least 2n − 2| F F v |.
In this paper, we further consider the constraints that were not covered by Fu’s result [6]. We consider the ndimensional folded hypercube with some faulty elements such that every node is incident to at least two fault-free links. We proved that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | when every node in F Q n is incident to at least two fault-free links, where | F F v | + | F F e | 2n − 4. References [1] Ahmed El-Amawy, Shahram Latifi, Properties and performance of folded hypercubes, IEEE Transactions on Parallel and Distributed Systems 2 (1) (1991) 31–42. [2] J.C. Bermond (Ed.), Interconnection networks, Discrete Applied Mathematics (Special Issue) 37–38 (1992). [3] L. Bhuyan, D.P. Agrawal, Generalized hypercubes and hyperbus structure for a computer network, IEEE Transactions on Computers 33 (1984) 323–333. [4] Z.Z. Du, J. Jin, M.J. Ma, J.M. Xu, Cycle embedding in hypercubes with faulty vertices and edges, Journal of University of Science and Technology of China 38 (9) (2008) 1020–1024. [5] A.H. Esfahanian, L.M. Ni, B.E. Sagan, The twisted n-cube with application to multiprocessing, IEEE Transactions on Computers 40 (1991) 88. [6] Jung-Sheng Fu, Fault-free cycles in folded hypercubes with more faulty elements, Information Processing Letters 108 (5) (2008) 261– 263. [7] S.Y. Hsieh, Some edge-fault-tolerant properties of the folded hypercube, Networks 51 (2) (2008) 92–101. [8] D.F. Hsu, Interconnection networks and algorithms, Networks (Special Issue) 23 (4) (1993). [9] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays, Trees, Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992. [10] M.J. Ma, J.M. Xu, Z.Z. Du, Edge-fault-tolerant hamiltonicity of folded hypercubes, Journal of University of Science and Technology of China 36 (3) (2007) 244–248. [11] F.P. Preparata, J. Vuillemin, The cube-connected cycles: A versatile network for parallel computation, Communication of the ACM 24 (1981) 300–309. [12] D. Wang, Embedding Hamiltonian cycles into folded hypercubes with faulty links, Journal of Parallel and Distributed Computing 61 (4) (2001) 545–564. [13] Junming Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, 2001. [14] J.M. Xu, M.J. Ma, Cycles in folded hypercubes, Applied Mathematics Letters 19 (2) (2006) 140–145. [15] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant properties of hypercubes and folded hypercubes, Australasian Journal of Combinatorics 35 (2006) 7–16.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
A further result on fault-free cycles in faulty folded hypercubes Sun-Yuan Hsieh a , Che-Nan Kuo a , Hsin-Hung Chou b,∗ a b
Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan Department of Information Management, Chang Jung Christian University, Tainan County, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 2 April 2009 Received in revised form 29 September 2009 Accepted 2 October 2009 Available online 8 October 2009 Communicated by A.A. Bertossi Keywords: Folded hypercubes Interconnection networks Cycle embedding Faulty elements
Folded hypercube is a well-known variation of the hypercube structure and can be constructed from a hypercube by adding a link to every pair of nodes with complementary addresses. Let F F v (respectively, F F e ) be the set of faulty nodes (respectively, faulty links) in an n-dimensional folded hypercube F Q n . Fu has showed that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In this paper, we further consider the constraints | F F v | + | F F e | 2n − 4 and | F F e | n that were not covered by Fu’s result. We obtain the same lower bound of the longest fault-free cycle length, 2n − 2| F F v |, under the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Design of interconnection networks (networks for short) is essential to developing the parallel or distributed systems. There various topologies of interconnection networks have been proposed. The interested readers may refer to [2,8,13] for extensive references. Among the proposed network topologies, the hypercube [3] 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 [9]. Some variants of the hypercube have been proposed in the literature [1,5,11]. One of the most popular variants is the folded hypercube, which is an extension of the hypercube and can be constructed by adding a link to every pair of nodes with complementary addresses. The folded hypercube has been shown to be able to im-
*
Corresponding author. E-mail addresses: [email protected] (S.-Y. Hsieh), [email protected] (C.-N. Kuo), [email protected] (H.-H. Chou). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.10.003
© 2009 Elsevier B.V.
All rights reserved.
prove the system’s performance over a regular hypercube in many measurements [1,12]. 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 [9]. Since faults may happen on both nodes and links in a network, it is practically meaningful and important to consider faulty networks. In this paper, we consider faulty folded hypercubes, i.e., folded hypercubes with faulty elements which include faulty nodes and faulty links. A node is fault-free if it is not faulty. A link is fault-free if the communication link between end-nodes is not faulty. A path (cycle) is fault-free if it contains neither faulty nodes nor faulty links. Previously, the problem of fault-tolerant embedding on an n-dimensional folded hypercube F Q n has been studied in [6,7,10,12,14,15]. Let F F v and F F e be the sets of faulty nodes and faulty links of F Q n , respectively. Fu [6] has showed that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In this paper, we further consider the constraints | F F v | + | F F e | 2n − 4
42
S.-Y. Hsieh et al. / Information Processing Letters 110 (2009) 41–43
and | F F e | n that were not covered by Fu’s result. We obtain the same lower bound of the longest fault-free cycle length, 2n − 2| F F v |, under the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. 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. Concluding remarks are given in Section 4. 2. Preliminaries In this paper, a network topology is represented by a simple undirected graph, which is loopless and without multiple edges. We denote the vertex set and the edge set of a graph G by V (G ) and E (G ), respectively. We use an unordered pair of vertices (u , v ) to denote an edge with end-vertices (or end-nodes) u and v. The end-vertices of an edge are said to be incident with the edge, and vice versa. Two vertices which are incident with a common edge are adjacent. Two adjacent vertices are called neighbors each other. Throughout this paper, the terms – network and graph, node and vertex, link and edge – are used interchangeable. A path, denoted by v 0 , v 1 , . . . , v k , is a sequence of distinct vertices v 0 , v 1 , . . . , v k in which any two consecutive vertices are adjacent. We call v 0 and v k the end-vertices (or end-nodes) of the path. 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 . An n-dimensional hypercube Q n is an undirected graph with 2n nodes which are labelled by n-bit binary strings . . . 0 to 11 . . . 1. Two nodes are joined by a link from 00 n
n
if and only if their labels differ by one bit. In particular, a link is called a dimension k link if it links two nodes that differ in the kth bit. An n-dimensional folded hypercube F Q n can be constructed from an n-dimensional hypercube by adding a link to every pair of nodes with complementary addresses, e.g., node x = b1 b2 . . . bn and node x¯ = b¯1 b¯2 . . . b¯n where b¯i represents 1 − b i , the 1’s complement of b i . Thus, F Q n has 2n−1 more links than a regular Q n . We call these extra links skips to distinguish them from regular links. Fig. 1 illustrates a 2-dimensional and a 3-dimensional folded hypercubes. In the following, we introduce some previous results that will be employed later. In an n-dimensional folded hypercube F Q n , let E i be the set of links on dimension i, for 1 i n, and let E s be the set of skips. Xu et al. [15] have shown the result that the folded hypercubes are edgetransitive, Lemma 1, as follows. Lemma 1. (See [15].) There is an automorphism σ of F Q n such that σ ( E i ) = E j for i , j ∈ {1, 2, . . . , n} ∪ {s}. It directly derives the following corollary. Corollary 1. F Q n − E i is isomorphic to Q n for i ∈ {1, 2, . . . , n} ∪ { s }.
Fig. 1. Graphs of F Q 2 and F Q 3 , in which complementary links are drawn by dashed lines.
In an n-dimensional faulty folded hypercube F Q n , let F F v and F F e be the sets of faulty nodes and faulty links of F Q n , respectively. On the problem of finding the lower bound of longest fault-free cycle in F Q n , Fu [6] has shown the result as Lemma 2. Lemma 2. (See [6].) F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if | F F v | + | F F e | 2n − 4 and | F F e | n − 1. In an n-dimensional faulty hypercube F e be the sets of faulty nodes and faulty spectively. On the problem of finding the longest fault-free cycle in Q n , Du et al. [4] result as Lemma 3.
Q n , let F v and links of Q n , relower bound of have shown the
Lemma 3. (See [4].) Q n − F v − F e for n 3 contains a fault-free cycle of length at least 2n − 2| F v | if (1) | F v | + | F e | 2n − 4 and | F e | 2n − 5 and (2) every node in Q n is incident to at least two fault-free links. 3. Fault-free cycle in the faulty folded hypercubes In this section, we present our main result on considering the constraints that (1) | F F v | + | F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links, as shown in Theorem 1. In an n-dimensional faulty folded hypercube F Q n , we call a node k-free if it is incident to at most k fault-free links. Lemma 4. For n 3, if | F F v | + | F F e | 2n − 4, there is at most one 2-free node contained in F Q n .
S.-Y. Hsieh et al. / Information Processing Letters 110 (2009) 41–43
43
Case 2. There is a unique 2-free node u in F Q n and every node in F Q n − {u } is k-free for some k 3. In this case, the number of faulty links incident to u is n − 1. We can choose a link dimension j ∈ {1, 2, . . . , n} ∪ {s} on which there exists at least one faulty link incident to u, such that F Q n − E j is isomorphic to a hypercube Q n . With the same arguments as Case 1, we have that F Q n − E j also satisfies the constraints in Lemma 3. It derives that F Q n − F F v − F F e contains a fault-free cycle of length at least 2n − 2| F F v |. By combining the above two cases, we complete the proof. 2 4. Concluding remarks
Fig. 2. Illustration the proof of Lemma 4.
Proof. Assume that F Q n contains two 2-free nodes u and v. Let us consider a special case that u and v are adjacent. (See Fig. 2.) The least total number of faulty elements is (n − 1) + (n − 1) − 1 = 2n − 3, which contradicts to the assumption that | F F v | + | F F e | 2n − 4. Thus, F Q n contains at most one 2-free node. 2 Theorem 1. F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | if (1) | F F v |+| F F e | 2n − 4 and (2) every node in F Q n is incident to at least two fault-free links. Proof. By Lemma 2, F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | for | F F v | + | F F e | 2n − 4 and | F F e | n − 1. Thus, we only need to consider the constraints | F F v | + | F F e | 2n − 4 and | F F e | n. Since | F F v | + | F F e | 2n − 4 and | F F e | n, we have that | F F v | n − 4. By Lemma 4, we have that F Q n contains at most one 2-free nodes when | F F v | + | F F e | 2n − 4. Then, we consider the following two cases according to the number of 2-free nodes: Case 1. F Q n contains no 2-free node. That is, every node in F Q n is k-free for some k 3. In this case, we can choose a link dimension j ∈ {1, 2, . . . , n} ∪ {s} on which there exists at least one faulty link. By Corollary 1, we have that F Q n − E j is isomorphic to a hypercube Q n . Let F v (respectively, F e ) be the set of faulty nodes (respectively, faulty links) in Q n . Then, | F v | = | F F v |, | F e | = | F F e − E j | 2n − 5 and | F v | + | F e | = | F F v | + | F e | 2n − 4. Since every node in F Q n is k-free for some k 3, every node in F Q n − E j is incident to at least two fault-free links. By Lemma 3, there exists a fault-free cycle of length at least 2n − 2| F v |(= 2n − 2| F F v |) in Q n − F v − F e . Since Q n − F v − F e ⊂ F Q n − F F v − F F e , we obtain that F Q n − F F v − F F e contains a fault-free cycle of length at least 2n − 2| F F v |.
In this paper, we further consider the constraints that were not covered by Fu’s result [6]. We consider the ndimensional folded hypercube with some faulty elements such that every node is incident to at least two fault-free links. We proved that F Q n − F F v − F F e for n 3 contains a fault-free cycle of length at least 2n − 2| F F v | when every node in F Q n is incident to at least two fault-free links, where | F F v | + | F F e | 2n − 4. References [1] Ahmed El-Amawy, Shahram Latifi, Properties and performance of folded hypercubes, IEEE Transactions on Parallel and Distributed Systems 2 (1) (1991) 31–42. [2] J.C. Bermond (Ed.), Interconnection networks, Discrete Applied Mathematics (Special Issue) 37–38 (1992). [3] L. Bhuyan, D.P. Agrawal, Generalized hypercubes and hyperbus structure for a computer network, IEEE Transactions on Computers 33 (1984) 323–333. [4] Z.Z. Du, J. Jin, M.J. Ma, J.M. Xu, Cycle embedding in hypercubes with faulty vertices and edges, Journal of University of Science and Technology of China 38 (9) (2008) 1020–1024. [5] A.H. Esfahanian, L.M. Ni, B.E. Sagan, The twisted n-cube with application to multiprocessing, IEEE Transactions on Computers 40 (1991) 88. [6] Jung-Sheng Fu, Fault-free cycles in folded hypercubes with more faulty elements, Information Processing Letters 108 (5) (2008) 261– 263. [7] S.Y. Hsieh, Some edge-fault-tolerant properties of the folded hypercube, Networks 51 (2) (2008) 92–101. [8] D.F. Hsu, Interconnection networks and algorithms, Networks (Special Issue) 23 (4) (1993). [9] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays, Trees, Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992. [10] M.J. Ma, J.M. Xu, Z.Z. Du, Edge-fault-tolerant hamiltonicity of folded hypercubes, Journal of University of Science and Technology of China 36 (3) (2007) 244–248. [11] F.P. Preparata, J. Vuillemin, The cube-connected cycles: A versatile network for parallel computation, Communication of the ACM 24 (1981) 300–309. [12] D. Wang, Embedding Hamiltonian cycles into folded hypercubes with faulty links, Journal of Parallel and Distributed Computing 61 (4) (2001) 545–564. [13] Junming Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, 2001. [14] J.M. Xu, M.J. Ma, Cycles in folded hypercubes, Applied Mathematics Letters 19 (2) (2006) 140–145. [15] J.M. Xu, M.J. Ma, Z.Z. Du, Edge-fault-tolerant properties of hypercubes and folded hypercubes, Australasian Journal of Combinatorics 35 (2006) 7–16.