Fault-tolerant strong Menger (edge) connectivity and 3-extra edge-connectivity of balanced hypercubes

Li, Pingshan; Xu, Min

doi:10.1016/j.tcs.2017.10.017

Cited by 49 publications

(10 citation statements)

References 24 publications

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“…In this section, we first review bubble-sort star graphs and give some notations which will be used in the following proof. Definition 2.1 [7] The n-dimensional bubble-sort star graph BS n has vertex set that consists of all n! permutations on {1, 2,…”

Section: Preliminariesmentioning

confidence: 99%

“…For example, Qiao et al [10] proved that folded hypercube is (2n − 2)-conditional edge-fault-tolerant strongly Menger edge connected. Li et al discussed the edge-fault-tolerant strong Menger edge connectivity of the hypercubelike network [6] and the balanced hypercube [7]. He et al [5] considered the strong Menger edge connectivity of the regular network.…”

Section: Introductionmentioning

confidence: 99%

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Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs

Guo¹

2019

Preprint

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The connectivity and edge connectivity of interconnection network determine the fault tolerance of the network. An interconnection network is usually viewed as a connected graph, where vertex corresponds processor and edge corresponds link between two distinct processors. Given a connected graph G with vertex set V (G) and edge set E(G), if for any two distinct verticespaths between u and v, then G is strongly Menger edge connected. Let m be an integer with m ≥ 1. If G − F e remains strongly Menger edge connected for any F e ⊆ E(G) with |F e | ≤ m, then G is m-edge-fault-tolerant strongly Menger edge connected. If G − F e is strongly Menger edge connected for any F e ⊆ E(G) with |F e | ≤ m and δ(G−F e ) ≥ 2, then G is m-conditional edge-fault-tolerant strongly Menger edge connected.In this paper, we consider the n-dimensional bubble-sort star graph BS n . We show that BS n is (2n − 5)-edge-fault-tolerant strongly Menger edge connected for n ≥ 3 and (6n − 17)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 4. Moreover, we give some examples to show that our results are optimal.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs

Guo¹

2019

Preprint

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“…Lü [6] showed that λ 2 (BH n ) = 6n − 4 for n ≥ 2. Li et al [5] and Yang et al [12] independently proved that λ 3 (BH n ) = 8n − 8 for n ≥ 2. In addition, Yang et al [12] proposed a conjecture about the g-extra edge-connectivity of BH n as follows.…”

Section: Introductionmentioning

confidence: 99%

The $g$-extra edge-connectivity of balanced hypercubes

Wei

Yang

2020

Preprint

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The g-extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464-473] determined the 3-extra edge-connectivity of balanced hypercubes BH n and conjectured that the g-extra edge-connectivity of BH n is λ g (BH n ) = 2(g + 1)n − 4g + 4 for 2 ≤ g ≤ 2n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 12 g + 1 and 2 ≤ g ≤ 8, and disprove their conjecture for n ≥ 3e g (BH n ) g + 1 and 9 ≤ g ≤ 2n − 1, where

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“…The study on strong Menger vertex/edge connectedness attracts more and more researchers' attention. On one hand, the (conditional) strong Menger connectivity of many known networks were explored in literature, for example, star graph [9,12], hypercubes [13], folded hypercubes [21], hypercube-like networks [16], Cayley graphs generated by transposition trees [17], augmented cubes [2], bubble-sort star graph [1], balanced hypercubes [7] etc. On the other hand, the (conditional) strong Menger edge connectivity were investigated on hypercubes [14], folded hypercubes [3], hypercube-like networks [8], balanced hypercubes [7] etc.…”

Section: Introductionmentioning

confidence: 99%

“…On one hand, the (conditional) strong Menger connectivity of many known networks were explored in literature, for example, star graph [9,12], hypercubes [13], folded hypercubes [21], hypercube-like networks [16], Cayley graphs generated by transposition trees [17], augmented cubes [2], bubble-sort star graph [1], balanced hypercubes [7] etc. On the other hand, the (conditional) strong Menger edge connectivity were investigated on hypercubes [14], folded hypercubes [3], hypercube-like networks [8], balanced hypercubes [7] etc. He et al [6] and Sabir and Meng [15] presented sufficient conditions for a regular graph to be strongly Menger vertex/edge connected with some restricted conditions.…”

Section: Introductionmentioning

confidence: 99%

Strong Menger connectedness of augmented $k$-ary $n$-cubes

Gu¹,

Chang²,

Hao³

2019

Preprint

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A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x, y of G, there are min{deg G (x), deg G (y)} vertex(edge)-disjoint paths between x and y. In this paper, we consider strong Menger (edge) connectedness of the augmented k-ary n-cube AQ n,k , which is a variant of k-ary n-cube Q k n . By exploring the topological proprieties of AQ n,k , we show that AQn,3 for n ≥ 4 (resp. AQ n,k for n ≥ 2 and k ≥ 4) is still strongly Menger connected even when there are 4n − 9 (resp. 4n − 8) faulty vertices and AQ n,k is still strongly Menger edge connected even when there are 4n − 4 faulty edges for n ≥ 2 and k ≥ 3. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that AQ n,k is still strongly Menger edge connected even when there are 8n − 10 faulty edges for n ≥ 2 and k ≥ 3. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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Fault-tolerant strong Menger (edge) connectivity and 3-extra edge-connectivity of balanced hypercubes

Cited by 49 publications

References 24 publications

Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs

Edge-fault-tolerant strong Menger edge connectivity of bubble-sort star graphs

The $g$-extra edge-connectivity of balanced hypercubes

Strong Menger connectedness of augmented $k$-ary $n$-cubes

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