Panconnectivity, fault‐tolerant hamiltonicity and hamiltonian‐connectivity in alternating group graphs

Chang, Jou–Ming; Yang, Jinn‐Shyong; Wang, Yue-Li; Cheng, Yuwen

doi:10.1002/net.20039

Cited by 76 publications

(58 citation statements)

References 30 publications

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“…Proof We prove this theorem by induction on n. The result clearly holds for n = 1 because Q 3 1 is a cycle of length 3. By Lemma 2, the result also holds for n = 2.…”

Section: Lengthmentioning

confidence: 92%

“…Lemma 2 For any two distinct nodes x, y ∈ V (Q 3 2 ) and any integer l with d(x, y) ≤ l ≤ 8, Q 3 2 contains an x-y path of length l.…”

Section: Lemma 1 [10] the K-ary N-cube Is Hamiltonian-connected When mentioning

confidence: 99%

“…That is, any path of possible length can be embedded into a panconnected graph with dilation one. 1 It is recently received much attention on the problem of finding paths of various lengths in interconnection networks [3,7,8] because this is an important measurement for determining if the topology of a network is suitable for an application in which mapping paths of various lengths into the topology is required.…”

Section: Introductionmentioning

confidence: 99%

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Panconnectivity and edge-pancyclicity of 3-ary N-cubes

2007

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“…Proof We prove this theorem by induction on n. The result clearly holds for n = 1 because Q 3 1 is a cycle of length 3. By Lemma 2, the result also holds for n = 2.…”

Section: Lengthmentioning

confidence: 92%

“…Lemma 2 For any two distinct nodes x, y ∈ V (Q 3 2 ) and any integer l with d(x, y) ≤ l ≤ 8, Q 3 2 contains an x-y path of length l.…”

Section: Lemma 1 [10] the K-ary N-cube Is Hamiltonian-connected When mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Panconnectivity and edge-pancyclicity of 3-ary N-cubes

2007

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“…The notions in the preceding paragraph have been investigated in the context of a number of interconnection networks: for example, in crossed cubes [12], [31], Mö bius cubes [14], augmented cubes [20], alternating group graphs [9], star graphs [29], bubble-sort graphs [17], and in hypercubes and hypercube-like networks [13], [19], [22], [25], [26], [28], [30]. With regard to k-ary n-cubes, these notions have been considered in [15] and [27].…”

Section: Introductionmentioning

confidence: 99%

Bipanconnectivity and Bipancyclicity in k-ary n-cubes

Stewart

Xiang

2009

IEEE Trans. Parallel Distrib. Syst.

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Publisher's copyright statement:This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. c 2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. is bipanconnected and edge-bipancyclic, when k ! 3 and n ! 2, and we also show that when k is odd, Q k n is m-panconnected, for m ¼ nðkÀ1Þþ2kÀ6 2, and ðk À 1Þ-pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening, then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q k n , even in the presence of a faulty processor.

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“…That is why many parallel algorithms have been devised based on them [13,14,16]. Many researchers have discussed how to embed cycles and paths into various interconnection networks [1,3,6,7,9,17,22,23]. A graph is Hamiltonian if it embeds a Hamiltonian cycle that contains each vertex exactly once [4].…”

mentioning

confidence: 99%

The panconnectivity and the pancycle-connectivity of the generalized base-b hypercube

Huang

Fang

2008

J Supercomput

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The interconnection network considered in this paper is the generalized base-b hypercube that is an attractive variant of the well-known hypercube. The generalized base-b hypercube is superior to the hypercube in many criteria, such as diameter, connectivity, and fault diameter. In this paper, we study the panconnectivity and pancycle-connectivity of the generalized base-b hypercube. We show that a generalized base-b hypercube is panconnected for b ≥ 3. That is, for each pair of distinct vertices x and y of the n-dimensional generalized base-b hypercube GH(b, n) and for any integer l, where Dist(x, y) ≤ l ≤ N − 1, there exists a path of the length l joining x and y, where N is the order of the graph GH(b, n) and Dist(x, y) is the distance between x and y. We also show that a generalized base-b hypercube is pancycleconnected for b ≥ 3. That is, every two distinct vertices x and y of the graph GH(b, n) are contained by a cycle of every length ranging from the length of the smallest cycle that contains x and y to N .

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Panconnectivity, fault‐tolerant hamiltonicity and hamiltonian‐connectivity in alternating group graphs

Cited by 76 publications

References 30 publications

Panconnectivity and edge-pancyclicity of 3-ary N-cubes

Panconnectivity and edge-pancyclicity of 3-ary N-cubes

Bipanconnectivity and Bipancyclicity in k-ary n-cubes

The panconnectivity and the pancycle-connectivity of the generalized base-b hypercube

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