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Some results on topological properties of folded hypercubes
Information Processing Letters 109 (2009) 395–399
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Information Processing Letters www.elsevier.com/locate/ipl
Some results on topological properties of folded hypercubes ✩ Xie-Bin Chen Department of Mathematics and Information Science, Zhangzhou Teachers College, Zhangzhou, Fujian 363000, China
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Article history: Received 16 August 2008 Received in revised form 7 December 2008 Accepted 8 December 2008 Available online 13 December 2008 Communicated by M. Yamashita Keywords: Folded hypercube Hypercube Fault-tolerance Panconnectivity Hamiltonicity Hamiltonian-laceability
An n-dimensional folded hypercube FQ n is an attractive variance of an n-dimensional hypercube Q n , it is obtained by adding an edge between every pair of vertices with complementary addresses. Recently, Hsieh studied edge-fault-tolerant Hamiltonicity of FQ n , and Fang studied (bi)panconnectivity of FQ n . In this paper, we first give a result on the connection between FQ n and Q n , then applying known topological properties of hypercubes, we improve the results of Hsieh; also, we obtain some results on fault-tolerant (bi)panconnectivity of FQ n that generalize the results of Fang. By our method it is possible to obtain other topological properties of FQ n . © 2008 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we follow [1] for graph-theoretical terminology and notation, and a graph G = ( V , E ) means a simple graph, where V = V (G ) is the vertex-set and E = E (G ) is the edge-set of the graph G. A graph P = v 0 v 1 . . . v k is called a path if k + 1 vertices v 0 , v 1 , . . . , v k are distinct and v i −1 v i is an edge of P for i = 1, 2, . . . , k, two vertices v 0 and v k are called end-vertices of the path P . The number k of the edges contained in a path P is called the length of P , denoted by l( P ). If x and y are two end-vertices of a path P , then P is said to be an (x, y )-path. The length of a shortest (x, y )-path in a graph G is called the distance between x and y in G, and denoted by d G (x, y ). Let D (G ) = max{d G (x, y ) | x, y ∈ V (G )}, D (G ) is called the diameter of the graph G. A graph C = v 1 v 2 . . . v k , where k 3, is called a cycle, if k vertices v 1 , v 2 , . . . , v k are distinct, v i v i +1 is an edge of C for i = 1, 2, . . . , k − 1, and v k v 1 is also an edge of C . The number k of the edges contained in a cycle C is called the length of C , denoted by l(C ). A path (resp. cycle) containing all vertices of a graph is called a Hamil-
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The work was supported by NNSF of China (No. 10671191). E-mail address: [email protected].
0020-0190/$ – see front matter doi:10.1016/j.ipl.2008.12.005
© 2008 Elsevier B.V.
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tonian path (resp. Hamiltonian cycle). A graph containing a Hamiltonian cycle is called Hamiltonian. A graph G is called Hamiltonian-connected if there exists an Hamiltonian path between its any two vertices. A bipartite graph G is called Hamiltonian-laceable [18] if there exists a Hamiltonian path between any two vertices of its different partite sets; a Hamiltonian-laceable graph G is called strongly Hamiltonian-laceable [10] if there exists a path of length | V (G )| − 2 between any two vertices of its same partite set; Hamiltonian-laceable graph G is called hyper-Hamiltonian-laceable [13] if for any vertex v there exists an Hamiltonian path of G − v between any two vertices of its partite set without v. A graph G is called m-panconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l with m l | V (G )| − 1, where m D (G ); moreover, an m-panconnected graph is called strictly m-panconnected if it is not (m − 1)-panconnected [6]. A graph G is called panconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l such that d G (x, y ) l | V (G )|− 1. A graph G is called bipanconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l such that d G (x, y ) l | V (G )| − 1 and l − d G (x, y ) is even. Notice that in [6,14] the bipanconnectivity was defined only for bipartite graphs.
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The n-dimensional hypercube or n-cube, denoted by Q n , is a bipartite graph with 2n vertices, its any vertex v is denoted by an n-bit binary string v = δn δn−1 . . . δ2 δ1 , where δi ∈ {0, 1} for all i , 1 i n. Two vertices of Q n are adjacent if and only if their binary strings differ in exactly one bit position, so Q n is an n-regular graph. The vertex whose every bit is 0 is denoted by O . The Hamming distance h(x, y ) between two vertices x and y in Q n is the number of different bits in the corresponding strings of the both vertices. Clearly, h(x, y ) is exactly the distance between x and y in Q n . If x = xn xn−1 . . . x2 x1 and y = yn yn−1 . . . y 2 y 1 are two vertices in Q n such that y i = 1 − xi for i = 1, 2, . . . , n, then we denote y = x¯ , and we say that x and x¯ have complementary addresses. Assume e = uv is an edge of Q n and the two binary strings of u and v differ in the ith bit, then e is called an edge of dimension i in Q n . The set of all edges of dimension i in Q n is denoted by E i , it is clear that E i is a perfect matching of Q n for all i , 1 i n. It is well known that the n-cube is one of the most popular and efficient interconnection networks due to its many excellent properties. Link and/or processor failures are inevitable when a large parallel computer system is put in use. Therefore, the faulttolerant capacity of a network is a critical issue in parallel computing. There is a large amount of literature on (faulttolerant) properties of hypercubes. See recent papers [2,4, 7,14–17,20–24]. As a variance of the n-cube Q n , the n-dimensional folded hypercube, denoted by FQ n (n 2), is defined as follows (see, e.g., [5]): FQ n is an (n + 1)-regular graph, its vertex set is exactly V ( Q n ), and its edge set is E ( Q n ) ∪ E 0 , where E 0 = {xx¯ | x ∈ V ( Q n )}. In other words, FQ n is a graph obtained from Q n by adding an edge, called a complementary edge, between any pair of vertices with complementary addresses. Clearly, E i is a perfect matching of FQ n for every i = 0, 1, . . . , n. The n-dimensional folded hypercube FQ n is superior to the n-cube Q n in some properties. For example, its diameter is n2 , about half the diameter of Q n [5]; there are n + 1 internally disjoint paths of length at most n2 + 1 between any pair of vertices [19]; there exists a cycle of every length l with n + 1 l 2n − 1 if n is even [24]. Many results on folded hypercubes can be found in papers [3,5,6,8,9,11,12,19,24–26]. Let ω be some topological property of graphs. A graph G is said to be ω if G has property ω ; G is said to be k(resp. k-vertex-, k-edge-) fault-tolerant ω if G − F is also ω, where F is any set with at most k vertices and/or edges (k vertices, k edges). For example, the n-cube Q n with n 2 is (n − 2)-edge-fault-tolerant strongly Hamiltonianlaceable [22]. Recently, Hsieh [9] studied edge-fault-tolerant Hamiltonicity of FQ n , Fang [6] studied (bi)panconnectivity of FQ n . In this paper, we first give a result on the connection between FQ n and Q n , then applying known topological properties of hypercubes, we improve the results of Hsieh; also, we obtain some results on fault-tolerant (bi)panconnectivity of FQ n that generalize the results of Fang. By our method it is possible to obtain other topological properties of FQ n .
2. Some lemmas In this section, to prove main results we give some lemmas. Lemma 1. (See [5].) Let x and y be any two vertices in FQ n , h(x, y ) denote their Hamming distance and d(x, y ) denote their distance in FQ n . Then d(x, y ) = h(x, y ) if h(x, y ) n2 , and d(x, y ) = n + 1 − h(x, y ) otherwise. Therefore, the diameter of FQ n is n2 . Lemma 2. (See [24].) In FQ n , assume that E i is the set of all edges of dimension i for i = 1, 2, . . . , n, and E 0 is the set of all complementary edges. Then there is an automorphism σ of FQ n such that σ ( E i ) = E j and FQ n − E i ∼ = Q n for all i , j ∈ {0, 1, . . . , n}. Lemma 3. (See [25].) (1) FQ n ia a bipartite graph if and only if n is odd. (2) If n is even, then the length of any shortest odd cycle in FQ n is n + 1. Lemma 4. (See [16].) Let x and y be any two vertices in Q n and h = h(x, y ) denote their Hamming distance. Then there exist n internally disjoint (x, y )-paths such that h of them are of length h, which lie in a h-dimensional subcube, and the others are of length h + 2. Especially, if x and y are any two vertices with complementary addresses, then there exist n internally disjoint (x, y )-paths of length n. Lemma 5. (See [22].) If n 3, then Q n is (n − 3)-edge-faulttolerant hyper Hamiltonian-laceable. Lemma 6. (See [20].) If n 3, then Q n is conditional (2n − 5)edge-fault-tolerant strongly Hamiltonian-laceable, where ‘conditional’ means that its any vertex is incident with at least two fault-free edges (a necessary condition for Hamiltonianlaceability). Lemma 7. (See [15].) Let F v and F e denote a set of faulty vertices and edges in Q n with n 2, respectively. Assume x and y are any two vertices in Q n − F v − F e . If | F v | + | F e | n − 2, then there exists a fault-free (x, y )-path of every length l such that h(x, y ) + 2 l 2n − 2| F v | − 1 and l − h(x, y ) is even, where h(x, y ) is their Hamming distance and is also their distance in Q n . 3. Main results In this section, Theorem 1 is a result on the connection between FQ n and Q n , Theorem 2 improves the results of Tsieh on edge-fault-tolerant Hamiltonian-connectivity and edge-fault-tolerant strongly (hyper-)Hamiltonian laceability of FQ n , Theorem 3 concerns fault-tolerant (bi)panconnectivity of FQ n , and Corollary 1 is the generalization of the results of Fang. Theorem 1. Let x and y be any two vertices in FQ n , h(x, y ) denote their Hamming distance, and d(x, y ) denote their distance
X.-B. Chen / Information Processing Letters 109 (2009) 395–399
in FQ n . Then there exist two n-cubes Q and Q contained in FQ n with the following properties: (1) There exist an (x, y )-path P of length h(x, y ) in Q and an (x, y )-path R of length n + 1 − h(x, y ) in Q such that C := P ∪ R is a cycle of length n + 1 in FQ n , and | E (C ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n. (2) d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ), or d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). (3) There exist an unique h(x, y )-cube W contained in the ncube Q such that P ⊂ W and an unique (n + 1 − h(x, y ))cube W contained in the n-cube Q such that R ⊂ W . Moreover, E ( W ) ∩ E ( W ) = ∅ and V ( W ) ∩ V ( W ) = {x, y }. (4) In FQ n there exist n + 1 internally disjoint (x, y )-paths such that h(x, y ) of them are of length h(x, y ) and the others are of length n + 1 − h(x, y ). Therefore, the connectivity of FQ n is n + 1. Proof. Let h = h(x, y ). (1) Firstly, we assume that x = O and y = δn δn−1 . . . δ2 δ1 such that δi = 1 for i = 1, 2, . . . , h and δ j = 0 for j = h + 1, h + 2, . . . , n. Let P = u 0 u 1 . . . u h be an (x, y )-path of length h in FQ n , where u 0 = x, u h = y and u i −1 u i ∈ E i for i = 1, 2, . . . , h. And let R = v 0 v 1 . . . v n+1−h be a ( y , x)path of length n + 1 − h in FQ n , where v 0 = y , v n+1−h = x, v n−h v n−h+1 ∈ E 0 and v i −1 v i ∈ E h+i for i = 1, 2, . . . , n − h. Since FQ n is vertex transitive, for any two vertices x and y with their Hamming distance h, we can similarly construct two (x, y )-paths P and R such that l( P ) = h and l( R ) = n + 1 − h, and C := P ∪ R is a cycle of length n + 1 and | E (C ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n. Assume E ( P ) ∩ E i = ∅ and E ( R ) ∩ E j = ∅, where i = j. Clearly, P is an (x, y )-path in Q := FQ n − E i and R is an (x, y )-path in Q := FQ n − E j . By Lemma 2, Q and Q are two n-cubes contained in FQ n . (2) Since P contains no edges of the same dimension, then d Q (x, y ) = h, similarly, d Q (x, y ) = n + 1 − h. If h n2 , by Lemma 1, then d(x, y ) = h, thus d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). If h > n2 , by Lemma 1, then d(x, y ) = n + 1 − h, thus d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). (3) Since l( P ) = h and P contains no edges of the same dimension, then there exists an unique h-cube W such that P ⊂ W ⊂ Q . Similarly, there exists an unique (n + 1 − h)-cube W such that R ⊂ W ⊂ Q . Since | E ( P ∪ R ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n, then E ( W ) ∩ E ( W ) = ∅. Clearly, {x, y } ⊂ V ( W ) ∩ V ( W ). Suppose that z = x, z = y and z ∈ V ( W ) ∩ V ( W ), we shall deduce a contrary. Since l1 := d W (x, z) < d W (x, y ) = h and l2 := d W (x, z) < d W (x, y ) = n + 1 − h, in W there exists a shortest (x, z)path P of length l1 with l1 h − 1, and in W there exists a shortest (x, z)-path R of length l2 with l2 n − h. Hence, C := P ∪ R is a cycle of length l = l1 + l2 n − 1 and contains no edges of the same dimension. Assume E (C ) ∩ E j = ∅ for some j, 0 j n, then C ⊂ (FQ n − E j ) ∼ = Q n. It is clear that the even cycle C of the n-cube Q n must contain edges of the same dimension, this is a contrary. Hence, V ( W ) ∩ V ( W ) = {x, y }. (4) Since W is an h-cube contained in FQ n , W is an (n + 1 − h)-cube contained in FQ n , V ( W ) ∩ V ( W ) =
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{x, y }, d W (x, y ) = h and d w (x, y ) = n + 1 − h, by Lemma 4, the conclusion follows. The proof of Theorem 1 is complete. 2 S.-Y. Hsieh [9] recently obtained the following results: (a) If n ( 2) is even, then FQ n is (n − 2)-edge-faulttolerant Hamiltonian-connected. (b) If n ( 3) is odd, then FQ n is (n − 1)-edge-faulttolerant strongly Hamiltonian-laceable. (c) If n ( 3) is odd, then FQ n is (n − 2)-edge-faulttolerant hyper Hamiltonian-laceable. The following theorem improves the results (a) and (b) of Hsieh. Theorem 2. Assume F e denotes a set of faulty edges in FQ n . (1) If n ( 4) is even, | F e | 2n − 5, and any vertex of FQ n − F e is incident with at least three edges, then FQ n − F e is Hamiltonian-connected, that is, FQ n is conditional (2n − 5)-edge-fault-tolerant Hamiltonian-connected when n ( 4) is even, where “conditional” means that its any vertex is incident with at least three fault-free edges (a necessary condition for Hamiltonian-connectivity). (2) If n ( 3) is odd, | F e | 2n − 4, and any vertex of FQ n − F e is incident with at least two edges, then FQ n − F e is strongly Hamiltonian-laceable, that is, FQ n is conditional (2n − 4)-edge-fault-tolerant strongly Hamiltonian-laceable when n ( 3) is odd, where “conditional” means that its any vertex is incident with at least two fault-free edges (a necessary condition for Hamiltonianlaceability). Proof. (1) Let x and y be any two vertices in FQ n , and d(x, y ) denote their distance in FQ n . By Theorem 1 (1) and (2), there exist two n-cubes Q and Q contained in FQ n such that d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). If d(x, y ) is odd, then d Q (x, y ) is odd; if d(x, y ) is even, since n is even, then d Q (x, y ) is odd. Hence, there exists an n-cube S contained in FQ n such that S = FQ n − E j for some j, 0 j n, and d S (x, y ) is odd, that is, x and y are in different partite sets in the n-cube S. Since any vertex in FQ n − F e is incident with at least three edges, then any vertex in S − F e is incident with at least two edges. Since | F e | 2n − 5, by Lemma 6, then in S − F e (⊂ FQ n − F e ) there exists an Hamiltonian path between x and y. Hence, FQ n is conditional (2n − 5)-edge-fault-tolerant Hamiltonian-connected when n ( 4) is even. (2) Since FQ n is n + 1-regular and | F e | 2n − 4, then in FQ n − F e there exists at most one vertex, say v, incident with exactly two edges. If any vertex of FQ n − F e is incident with at least three edges, then select any f ∈ F e , otherwise, select such an f ∈ F e that is incident with v in FQ n . Assume f ∈ E j for some j , 0 j n. Let S := FQ n − E j , then S is an n-cube, | E ( S ) ∩ F e | 2n − 5 and any vertex in S − F e is incident with at least two edges. By Lemma 6, S − F e is strongly Hamiltonian-laceable. Since n is odd, by Lemma 3, FQ n is a bipartite graph, Since S − F e ⊂ FQ n − F e ,
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by the definition of strongly Hamiltonian-laceability, the conclusion of (2) follows. The proof of Theorem 2 is complete. 2 Remark 1. Let n ( 3) be odd and F e ⊂ E (FQ n ) with 1 | F e | n − 2. Select any f ∈ F e , assume f ∈ E j for some j , 0 j n. Let S = FQ n − E j ∼ = Q n . Then | E ( S ) ∩ F e | n − 3. Since S − F e ⊂ FQ n − F e and FQ n is a bipartite graph, by Lemma 5 and the definition of hyperHamiltonian-laceability, the result (c) of Hsieh follows. J.-F. Fang [6] recently obtained the following results: (d) If n ( 3) is odd, then FQ n is bipanconnected. (e) If n ( 2) is even, then FQ n is strictly (n − 1)panconnected. We obtain the results on fault-tolerant (bi)panconnectivity and m-panconnectivity of folded hypercubes as follows. Theorem 3. Let F v and F e denote a set of faulty vertices and edges in FQ n , respectively, such that | F v | + | F e | n − 2. And let x and y be any two vertices in FQ n − F v − F e , and d(x, y ) denote their distance in FQ n . (1) If n is any integer with n 2, then in FQ n − F v − F e there exists an (x, y )-path of every length l such that d(x, y ) + 2 l 2n − 2| F v | − 1 and l − d(x, y ) is even. (2) If n ( 2) is even, then in FQ n − F v − F e there exists an (x, y )-path of every length l such that n − 1 l 2n − 2| F v | − 1. Moreover, the lower bound n − 1 of l and the upper bound n − 2 of | F e | are tight. Proof. Clearly, Theorem 3 holds for FQ 2 , we assume n 3 below. (1) By Theorem 1(2), there is an n-cube S contained in FQ n such that d S (x, y ) = d(x, y ). Since | F v | + | F e | n − 2, by Lemma 7, in S − F v − F e (⊂ FQ n − F v − F e ) there exists an (x, y )-path of every length l such that d(x, y ) + 2 = d S (x, y ) + 2 l 2n − 2| F v | − 1 and l − d(x, y ) is even. (2) Assume n ( 4) is even. By Lemma 1, d(x, y ) + 2 n/2 + 2 n. By Theorem 1(2), there is an n-cube S contained in FQ n such that d S (x, y ) = n + 1 − d(x, y ). Since | F v |+| F e | n − 2, by Lemma 7, in S − F v − F e (⊂ FQ n − F v − F e ) there exists an (x, y )-path of every length l such that n + 3 − d(x, y ) = d S (x, y ) + 2 l 2n − 2| F v | − 1 and l − (n + 3 − d(x, y )) is even. If d(x, y ) = 2, by Lemma 1, then h(x, y ) = 2 or h(x, y ) = n − 1. Since | F v | + | F e | n − 2, by Theorem 1(4), in FQ n − F v − F e there is an (x, y )-path of length n − 1. Similarly, if d(x, y ) = 1, then in FQ n − F v − F e there is an (x, y )-path of length n. We consider two cases. Case 1: d(x, y ) is even. If d(x, y ) = 2, then in S − F v − F e there exists an (x, y )path of every odd length l with n + 1 = n + 3 − 2 l 2n − 2| F v |− 1 and an (x, y )-path of length n − 1. If d(x, y ) 4 is even, then in S − F v − F e there exists an (x, y )-path of every odd length l with n − 1 = n + 3 − 4 l 2n −
2| F v | − 1. Note that d(x, y ) + 2 n, by (1) in FQ n − F v − F e there exists also an (x, y )-path of every even length l with n l 2n − 2| F v | − 2. Hence, in FQ n − F v − F e there exists an (x, y )-path of every length l with n − 1 l 2n − 2| F v | − 1. Case 2: d(x, y ) is odd. If d(x, y ) = 1, then in S − F v − F e there exists an (x, y )path of every even length l such that n + 2 = n + 3 − 1 l 2n − 2| F v | − 2 and an (x, y )-path of length n. If d(x, y ) 3 is odd, then in S − F v − F e there exists an (x, y )-path of every even length l such that n = n + 3 − 3 l 2n − 2| F v | − 2. Note that d(x, y ) + 2 n − 1, by (1) in FQ n − F v − F e there exists also an (x, y )-path of every odd length l such that n − 1 l 2n − 2| F v | − 1. Hence, in FQ n − F v − F e there exists an (x, y )-path of every length l such that n − 1 l 2n − 2| F v | − 1. When d(x, y ) = 1, by Lemma 3 there exists no (x, y )path of length n − 2, it follows that the lower bound n − 1 of l is tight. Assume that F e is incident with some vertex v in FQ n such that | F e | = n − 1. Then in FQ n − F e the vertex v is adjacent to exactly two vertices, denoted by x and y. Clearly, in FQ n − F e there is no Hamiltonian path between x and y. Hence, the upper bound n − 2 of | F e | is tight. The proof of Theorem 3 is complete. 2 The following corollary is a generalization of the results of Fang. Corollary 1. (1) If n is any integer with n 2, then FQ n is bipanconnected. (2) If n ( 2) is even, then FQ n is (n − 2)-edge-fault-tolerant strictly (n − 1)-panconnected, and the number n − 2 of tolerable faulty edges is tight. Proof. (1) Let F v = ∅ and F e = ∅ in Theorem 3(1), by the definition of bipanconnectivity, the conclusion follows. (2) Let F v = ∅ in Theorem 3(2), by the definition of strictly (n − 1)-panconnectivity, the conclusion follows. 2 Acknowledgements The author would like to express his gratitude to the anonymous referees for their kind suggestions and corrections that helped improve the original manuscript. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan Press, London, 1976. [2] X.-B. Chen, Cycles passing through prescribed edges in a hypercube with some faulty edges, Inform. Process. Lett. 104 (2007) 211–215. [3] S.A. Choudum, R.U. Nandini, Complete binary trees in folded and enhanced cubes, Networks 43 (2004) 266–272. [4] T. Dvoˇrák, Hamiltonian cycles with prescribed edges in hypercubes, SIAM J. Discrete Math. 19 (2005) 135–144. [5] A. El-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. Parallel Distrib. Syst. 2 (1991) 31–42. [6] J.-F. Fang, The bipanconnectivity and m-panconnectivity of the folded hypercube, Theoret. Comput. Sci. 385 (2007) 286–300. [7] F. Harary, J.P. Hayes, H.-J. Wu, A survey of the theory of hypercube graphs, Comput. Math. Appl. 15 (1988) 277–289.
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Some results on topological properties of folded hypercubes ✩ Xie-Bin Chen Department of Mathematics and Information Science, Zhangzhou Teachers College, Zhangzhou, Fujian 363000, China
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Article history: Received 16 August 2008 Received in revised form 7 December 2008 Accepted 8 December 2008 Available online 13 December 2008 Communicated by M. Yamashita Keywords: Folded hypercube Hypercube Fault-tolerance Panconnectivity Hamiltonicity Hamiltonian-laceability
An n-dimensional folded hypercube FQ n is an attractive variance of an n-dimensional hypercube Q n , it is obtained by adding an edge between every pair of vertices with complementary addresses. Recently, Hsieh studied edge-fault-tolerant Hamiltonicity of FQ n , and Fang studied (bi)panconnectivity of FQ n . In this paper, we first give a result on the connection between FQ n and Q n , then applying known topological properties of hypercubes, we improve the results of Hsieh; also, we obtain some results on fault-tolerant (bi)panconnectivity of FQ n that generalize the results of Fang. By our method it is possible to obtain other topological properties of FQ n . © 2008 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we follow [1] for graph-theoretical terminology and notation, and a graph G = ( V , E ) means a simple graph, where V = V (G ) is the vertex-set and E = E (G ) is the edge-set of the graph G. A graph P = v 0 v 1 . . . v k is called a path if k + 1 vertices v 0 , v 1 , . . . , v k are distinct and v i −1 v i is an edge of P for i = 1, 2, . . . , k, two vertices v 0 and v k are called end-vertices of the path P . The number k of the edges contained in a path P is called the length of P , denoted by l( P ). If x and y are two end-vertices of a path P , then P is said to be an (x, y )-path. The length of a shortest (x, y )-path in a graph G is called the distance between x and y in G, and denoted by d G (x, y ). Let D (G ) = max{d G (x, y ) | x, y ∈ V (G )}, D (G ) is called the diameter of the graph G. A graph C = v 1 v 2 . . . v k , where k 3, is called a cycle, if k vertices v 1 , v 2 , . . . , v k are distinct, v i v i +1 is an edge of C for i = 1, 2, . . . , k − 1, and v k v 1 is also an edge of C . The number k of the edges contained in a cycle C is called the length of C , denoted by l(C ). A path (resp. cycle) containing all vertices of a graph is called a Hamil-
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The work was supported by NNSF of China (No. 10671191). E-mail address: [email protected].
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© 2008 Elsevier B.V.
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tonian path (resp. Hamiltonian cycle). A graph containing a Hamiltonian cycle is called Hamiltonian. A graph G is called Hamiltonian-connected if there exists an Hamiltonian path between its any two vertices. A bipartite graph G is called Hamiltonian-laceable [18] if there exists a Hamiltonian path between any two vertices of its different partite sets; a Hamiltonian-laceable graph G is called strongly Hamiltonian-laceable [10] if there exists a path of length | V (G )| − 2 between any two vertices of its same partite set; Hamiltonian-laceable graph G is called hyper-Hamiltonian-laceable [13] if for any vertex v there exists an Hamiltonian path of G − v between any two vertices of its partite set without v. A graph G is called m-panconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l with m l | V (G )| − 1, where m D (G ); moreover, an m-panconnected graph is called strictly m-panconnected if it is not (m − 1)-panconnected [6]. A graph G is called panconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l such that d G (x, y ) l | V (G )|− 1. A graph G is called bipanconnected if for its any two vertices x and y, there exists an (x, y )-path of every length l such that d G (x, y ) l | V (G )| − 1 and l − d G (x, y ) is even. Notice that in [6,14] the bipanconnectivity was defined only for bipartite graphs.
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The n-dimensional hypercube or n-cube, denoted by Q n , is a bipartite graph with 2n vertices, its any vertex v is denoted by an n-bit binary string v = δn δn−1 . . . δ2 δ1 , where δi ∈ {0, 1} for all i , 1 i n. Two vertices of Q n are adjacent if and only if their binary strings differ in exactly one bit position, so Q n is an n-regular graph. The vertex whose every bit is 0 is denoted by O . The Hamming distance h(x, y ) between two vertices x and y in Q n is the number of different bits in the corresponding strings of the both vertices. Clearly, h(x, y ) is exactly the distance between x and y in Q n . If x = xn xn−1 . . . x2 x1 and y = yn yn−1 . . . y 2 y 1 are two vertices in Q n such that y i = 1 − xi for i = 1, 2, . . . , n, then we denote y = x¯ , and we say that x and x¯ have complementary addresses. Assume e = uv is an edge of Q n and the two binary strings of u and v differ in the ith bit, then e is called an edge of dimension i in Q n . The set of all edges of dimension i in Q n is denoted by E i , it is clear that E i is a perfect matching of Q n for all i , 1 i n. It is well known that the n-cube is one of the most popular and efficient interconnection networks due to its many excellent properties. Link and/or processor failures are inevitable when a large parallel computer system is put in use. Therefore, the faulttolerant capacity of a network is a critical issue in parallel computing. There is a large amount of literature on (faulttolerant) properties of hypercubes. See recent papers [2,4, 7,14–17,20–24]. As a variance of the n-cube Q n , the n-dimensional folded hypercube, denoted by FQ n (n 2), is defined as follows (see, e.g., [5]): FQ n is an (n + 1)-regular graph, its vertex set is exactly V ( Q n ), and its edge set is E ( Q n ) ∪ E 0 , where E 0 = {xx¯ | x ∈ V ( Q n )}. In other words, FQ n is a graph obtained from Q n by adding an edge, called a complementary edge, between any pair of vertices with complementary addresses. Clearly, E i is a perfect matching of FQ n for every i = 0, 1, . . . , n. The n-dimensional folded hypercube FQ n is superior to the n-cube Q n in some properties. For example, its diameter is n2 , about half the diameter of Q n [5]; there are n + 1 internally disjoint paths of length at most n2 + 1 between any pair of vertices [19]; there exists a cycle of every length l with n + 1 l 2n − 1 if n is even [24]. Many results on folded hypercubes can be found in papers [3,5,6,8,9,11,12,19,24–26]. Let ω be some topological property of graphs. A graph G is said to be ω if G has property ω ; G is said to be k(resp. k-vertex-, k-edge-) fault-tolerant ω if G − F is also ω, where F is any set with at most k vertices and/or edges (k vertices, k edges). For example, the n-cube Q n with n 2 is (n − 2)-edge-fault-tolerant strongly Hamiltonianlaceable [22]. Recently, Hsieh [9] studied edge-fault-tolerant Hamiltonicity of FQ n , Fang [6] studied (bi)panconnectivity of FQ n . In this paper, we first give a result on the connection between FQ n and Q n , then applying known topological properties of hypercubes, we improve the results of Hsieh; also, we obtain some results on fault-tolerant (bi)panconnectivity of FQ n that generalize the results of Fang. By our method it is possible to obtain other topological properties of FQ n .
2. Some lemmas In this section, to prove main results we give some lemmas. Lemma 1. (See [5].) Let x and y be any two vertices in FQ n , h(x, y ) denote their Hamming distance and d(x, y ) denote their distance in FQ n . Then d(x, y ) = h(x, y ) if h(x, y ) n2 , and d(x, y ) = n + 1 − h(x, y ) otherwise. Therefore, the diameter of FQ n is n2 . Lemma 2. (See [24].) In FQ n , assume that E i is the set of all edges of dimension i for i = 1, 2, . . . , n, and E 0 is the set of all complementary edges. Then there is an automorphism σ of FQ n such that σ ( E i ) = E j and FQ n − E i ∼ = Q n for all i , j ∈ {0, 1, . . . , n}. Lemma 3. (See [25].) (1) FQ n ia a bipartite graph if and only if n is odd. (2) If n is even, then the length of any shortest odd cycle in FQ n is n + 1. Lemma 4. (See [16].) Let x and y be any two vertices in Q n and h = h(x, y ) denote their Hamming distance. Then there exist n internally disjoint (x, y )-paths such that h of them are of length h, which lie in a h-dimensional subcube, and the others are of length h + 2. Especially, if x and y are any two vertices with complementary addresses, then there exist n internally disjoint (x, y )-paths of length n. Lemma 5. (See [22].) If n 3, then Q n is (n − 3)-edge-faulttolerant hyper Hamiltonian-laceable. Lemma 6. (See [20].) If n 3, then Q n is conditional (2n − 5)edge-fault-tolerant strongly Hamiltonian-laceable, where ‘conditional’ means that its any vertex is incident with at least two fault-free edges (a necessary condition for Hamiltonianlaceability). Lemma 7. (See [15].) Let F v and F e denote a set of faulty vertices and edges in Q n with n 2, respectively. Assume x and y are any two vertices in Q n − F v − F e . If | F v | + | F e | n − 2, then there exists a fault-free (x, y )-path of every length l such that h(x, y ) + 2 l 2n − 2| F v | − 1 and l − h(x, y ) is even, where h(x, y ) is their Hamming distance and is also their distance in Q n . 3. Main results In this section, Theorem 1 is a result on the connection between FQ n and Q n , Theorem 2 improves the results of Tsieh on edge-fault-tolerant Hamiltonian-connectivity and edge-fault-tolerant strongly (hyper-)Hamiltonian laceability of FQ n , Theorem 3 concerns fault-tolerant (bi)panconnectivity of FQ n , and Corollary 1 is the generalization of the results of Fang. Theorem 1. Let x and y be any two vertices in FQ n , h(x, y ) denote their Hamming distance, and d(x, y ) denote their distance
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in FQ n . Then there exist two n-cubes Q and Q contained in FQ n with the following properties: (1) There exist an (x, y )-path P of length h(x, y ) in Q and an (x, y )-path R of length n + 1 − h(x, y ) in Q such that C := P ∪ R is a cycle of length n + 1 in FQ n , and | E (C ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n. (2) d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ), or d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). (3) There exist an unique h(x, y )-cube W contained in the ncube Q such that P ⊂ W and an unique (n + 1 − h(x, y ))cube W contained in the n-cube Q such that R ⊂ W . Moreover, E ( W ) ∩ E ( W ) = ∅ and V ( W ) ∩ V ( W ) = {x, y }. (4) In FQ n there exist n + 1 internally disjoint (x, y )-paths such that h(x, y ) of them are of length h(x, y ) and the others are of length n + 1 − h(x, y ). Therefore, the connectivity of FQ n is n + 1. Proof. Let h = h(x, y ). (1) Firstly, we assume that x = O and y = δn δn−1 . . . δ2 δ1 such that δi = 1 for i = 1, 2, . . . , h and δ j = 0 for j = h + 1, h + 2, . . . , n. Let P = u 0 u 1 . . . u h be an (x, y )-path of length h in FQ n , where u 0 = x, u h = y and u i −1 u i ∈ E i for i = 1, 2, . . . , h. And let R = v 0 v 1 . . . v n+1−h be a ( y , x)path of length n + 1 − h in FQ n , where v 0 = y , v n+1−h = x, v n−h v n−h+1 ∈ E 0 and v i −1 v i ∈ E h+i for i = 1, 2, . . . , n − h. Since FQ n is vertex transitive, for any two vertices x and y with their Hamming distance h, we can similarly construct two (x, y )-paths P and R such that l( P ) = h and l( R ) = n + 1 − h, and C := P ∪ R is a cycle of length n + 1 and | E (C ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n. Assume E ( P ) ∩ E i = ∅ and E ( R ) ∩ E j = ∅, where i = j. Clearly, P is an (x, y )-path in Q := FQ n − E i and R is an (x, y )-path in Q := FQ n − E j . By Lemma 2, Q and Q are two n-cubes contained in FQ n . (2) Since P contains no edges of the same dimension, then d Q (x, y ) = h, similarly, d Q (x, y ) = n + 1 − h. If h n2 , by Lemma 1, then d(x, y ) = h, thus d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). If h > n2 , by Lemma 1, then d(x, y ) = n + 1 − h, thus d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). (3) Since l( P ) = h and P contains no edges of the same dimension, then there exists an unique h-cube W such that P ⊂ W ⊂ Q . Similarly, there exists an unique (n + 1 − h)-cube W such that R ⊂ W ⊂ Q . Since | E ( P ∪ R ) ∩ E j | = 1 for every j , j = 0, 1, . . . , n, then E ( W ) ∩ E ( W ) = ∅. Clearly, {x, y } ⊂ V ( W ) ∩ V ( W ). Suppose that z = x, z = y and z ∈ V ( W ) ∩ V ( W ), we shall deduce a contrary. Since l1 := d W (x, z) < d W (x, y ) = h and l2 := d W (x, z) < d W (x, y ) = n + 1 − h, in W there exists a shortest (x, z)path P of length l1 with l1 h − 1, and in W there exists a shortest (x, z)-path R of length l2 with l2 n − h. Hence, C := P ∪ R is a cycle of length l = l1 + l2 n − 1 and contains no edges of the same dimension. Assume E (C ) ∩ E j = ∅ for some j, 0 j n, then C ⊂ (FQ n − E j ) ∼ = Q n. It is clear that the even cycle C of the n-cube Q n must contain edges of the same dimension, this is a contrary. Hence, V ( W ) ∩ V ( W ) = {x, y }. (4) Since W is an h-cube contained in FQ n , W is an (n + 1 − h)-cube contained in FQ n , V ( W ) ∩ V ( W ) =
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{x, y }, d W (x, y ) = h and d w (x, y ) = n + 1 − h, by Lemma 4, the conclusion follows. The proof of Theorem 1 is complete. 2 S.-Y. Hsieh [9] recently obtained the following results: (a) If n ( 2) is even, then FQ n is (n − 2)-edge-faulttolerant Hamiltonian-connected. (b) If n ( 3) is odd, then FQ n is (n − 1)-edge-faulttolerant strongly Hamiltonian-laceable. (c) If n ( 3) is odd, then FQ n is (n − 2)-edge-faulttolerant hyper Hamiltonian-laceable. The following theorem improves the results (a) and (b) of Hsieh. Theorem 2. Assume F e denotes a set of faulty edges in FQ n . (1) If n ( 4) is even, | F e | 2n − 5, and any vertex of FQ n − F e is incident with at least three edges, then FQ n − F e is Hamiltonian-connected, that is, FQ n is conditional (2n − 5)-edge-fault-tolerant Hamiltonian-connected when n ( 4) is even, where “conditional” means that its any vertex is incident with at least three fault-free edges (a necessary condition for Hamiltonian-connectivity). (2) If n ( 3) is odd, | F e | 2n − 4, and any vertex of FQ n − F e is incident with at least two edges, then FQ n − F e is strongly Hamiltonian-laceable, that is, FQ n is conditional (2n − 4)-edge-fault-tolerant strongly Hamiltonian-laceable when n ( 3) is odd, where “conditional” means that its any vertex is incident with at least two fault-free edges (a necessary condition for Hamiltonianlaceability). Proof. (1) Let x and y be any two vertices in FQ n , and d(x, y ) denote their distance in FQ n . By Theorem 1 (1) and (2), there exist two n-cubes Q and Q contained in FQ n such that d Q (x, y ) = d(x, y ) and d Q (x, y ) = n + 1 − d(x, y ). If d(x, y ) is odd, then d Q (x, y ) is odd; if d(x, y ) is even, since n is even, then d Q (x, y ) is odd. Hence, there exists an n-cube S contained in FQ n such that S = FQ n − E j for some j, 0 j n, and d S (x, y ) is odd, that is, x and y are in different partite sets in the n-cube S. Since any vertex in FQ n − F e is incident with at least three edges, then any vertex in S − F e is incident with at least two edges. Since | F e | 2n − 5, by Lemma 6, then in S − F e (⊂ FQ n − F e ) there exists an Hamiltonian path between x and y. Hence, FQ n is conditional (2n − 5)-edge-fault-tolerant Hamiltonian-connected when n ( 4) is even. (2) Since FQ n is n + 1-regular and | F e | 2n − 4, then in FQ n − F e there exists at most one vertex, say v, incident with exactly two edges. If any vertex of FQ n − F e is incident with at least three edges, then select any f ∈ F e , otherwise, select such an f ∈ F e that is incident with v in FQ n . Assume f ∈ E j for some j , 0 j n. Let S := FQ n − E j , then S is an n-cube, | E ( S ) ∩ F e | 2n − 5 and any vertex in S − F e is incident with at least two edges. By Lemma 6, S − F e is strongly Hamiltonian-laceable. Since n is odd, by Lemma 3, FQ n is a bipartite graph, Since S − F e ⊂ FQ n − F e ,
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by the definition of strongly Hamiltonian-laceability, the conclusion of (2) follows. The proof of Theorem 2 is complete. 2 Remark 1. Let n ( 3) be odd and F e ⊂ E (FQ n ) with 1 | F e | n − 2. Select any f ∈ F e , assume f ∈ E j for some j , 0 j n. Let S = FQ n − E j ∼ = Q n . Then | E ( S ) ∩ F e | n − 3. Since S − F e ⊂ FQ n − F e and FQ n is a bipartite graph, by Lemma 5 and the definition of hyperHamiltonian-laceability, the result (c) of Hsieh follows. J.-F. Fang [6] recently obtained the following results: (d) If n ( 3) is odd, then FQ n is bipanconnected. (e) If n ( 2) is even, then FQ n is strictly (n − 1)panconnected. We obtain the results on fault-tolerant (bi)panconnectivity and m-panconnectivity of folded hypercubes as follows. Theorem 3. Let F v and F e denote a set of faulty vertices and edges in FQ n , respectively, such that | F v | + | F e | n − 2. And let x and y be any two vertices in FQ n − F v − F e , and d(x, y ) denote their distance in FQ n . (1) If n is any integer with n 2, then in FQ n − F v − F e there exists an (x, y )-path of every length l such that d(x, y ) + 2 l 2n − 2| F v | − 1 and l − d(x, y ) is even. (2) If n ( 2) is even, then in FQ n − F v − F e there exists an (x, y )-path of every length l such that n − 1 l 2n − 2| F v | − 1. Moreover, the lower bound n − 1 of l and the upper bound n − 2 of | F e | are tight. Proof. Clearly, Theorem 3 holds for FQ 2 , we assume n 3 below. (1) By Theorem 1(2), there is an n-cube S contained in FQ n such that d S (x, y ) = d(x, y ). Since | F v | + | F e | n − 2, by Lemma 7, in S − F v − F e (⊂ FQ n − F v − F e ) there exists an (x, y )-path of every length l such that d(x, y ) + 2 = d S (x, y ) + 2 l 2n − 2| F v | − 1 and l − d(x, y ) is even. (2) Assume n ( 4) is even. By Lemma 1, d(x, y ) + 2 n/2 + 2 n. By Theorem 1(2), there is an n-cube S contained in FQ n such that d S (x, y ) = n + 1 − d(x, y ). Since | F v |+| F e | n − 2, by Lemma 7, in S − F v − F e (⊂ FQ n − F v − F e ) there exists an (x, y )-path of every length l such that n + 3 − d(x, y ) = d S (x, y ) + 2 l 2n − 2| F v | − 1 and l − (n + 3 − d(x, y )) is even. If d(x, y ) = 2, by Lemma 1, then h(x, y ) = 2 or h(x, y ) = n − 1. Since | F v | + | F e | n − 2, by Theorem 1(4), in FQ n − F v − F e there is an (x, y )-path of length n − 1. Similarly, if d(x, y ) = 1, then in FQ n − F v − F e there is an (x, y )-path of length n. We consider two cases. Case 1: d(x, y ) is even. If d(x, y ) = 2, then in S − F v − F e there exists an (x, y )path of every odd length l with n + 1 = n + 3 − 2 l 2n − 2| F v |− 1 and an (x, y )-path of length n − 1. If d(x, y ) 4 is even, then in S − F v − F e there exists an (x, y )-path of every odd length l with n − 1 = n + 3 − 4 l 2n −
2| F v | − 1. Note that d(x, y ) + 2 n, by (1) in FQ n − F v − F e there exists also an (x, y )-path of every even length l with n l 2n − 2| F v | − 2. Hence, in FQ n − F v − F e there exists an (x, y )-path of every length l with n − 1 l 2n − 2| F v | − 1. Case 2: d(x, y ) is odd. If d(x, y ) = 1, then in S − F v − F e there exists an (x, y )path of every even length l such that n + 2 = n + 3 − 1 l 2n − 2| F v | − 2 and an (x, y )-path of length n. If d(x, y ) 3 is odd, then in S − F v − F e there exists an (x, y )-path of every even length l such that n = n + 3 − 3 l 2n − 2| F v | − 2. Note that d(x, y ) + 2 n − 1, by (1) in FQ n − F v − F e there exists also an (x, y )-path of every odd length l such that n − 1 l 2n − 2| F v | − 1. Hence, in FQ n − F v − F e there exists an (x, y )-path of every length l such that n − 1 l 2n − 2| F v | − 1. When d(x, y ) = 1, by Lemma 3 there exists no (x, y )path of length n − 2, it follows that the lower bound n − 1 of l is tight. Assume that F e is incident with some vertex v in FQ n such that | F e | = n − 1. Then in FQ n − F e the vertex v is adjacent to exactly two vertices, denoted by x and y. Clearly, in FQ n − F e there is no Hamiltonian path between x and y. Hence, the upper bound n − 2 of | F e | is tight. The proof of Theorem 3 is complete. 2 The following corollary is a generalization of the results of Fang. Corollary 1. (1) If n is any integer with n 2, then FQ n is bipanconnected. (2) If n ( 2) is even, then FQ n is (n − 2)-edge-fault-tolerant strictly (n − 1)-panconnected, and the number n − 2 of tolerable faulty edges is tight. Proof. (1) Let F v = ∅ and F e = ∅ in Theorem 3(1), by the definition of bipanconnectivity, the conclusion follows. (2) Let F v = ∅ in Theorem 3(2), by the definition of strictly (n − 1)-panconnectivity, the conclusion follows. 2 Acknowledgements The author would like to express his gratitude to the anonymous referees for their kind suggestions and corrections that helped improve the original manuscript. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan Press, London, 1976. [2] X.-B. Chen, Cycles passing through prescribed edges in a hypercube with some faulty edges, Inform. Process. Lett. 104 (2007) 211–215. [3] S.A. Choudum, R.U. Nandini, Complete binary trees in folded and enhanced cubes, Networks 43 (2004) 266–272. [4] T. Dvoˇrák, Hamiltonian cycles with prescribed edges in hypercubes, SIAM J. Discrete Math. 19 (2005) 135–144. [5] A. El-Amawy, S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. Parallel Distrib. Syst. 2 (1991) 31–42. [6] J.-F. Fang, The bipanconnectivity and m-panconnectivity of the folded hypercube, Theoret. Comput. Sci. 385 (2007) 286–300. [7] F. Harary, J.P. Hayes, H.-J. Wu, A survey of the theory of hypercube graphs, Comput. Math. Appl. 15 (1988) 277–289.
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