On embedding cycles into faulty twisted cubes

Yang, Ming-Chien; Li, Tseng-Kuei; Tan, Jinglu; Hsu, Lih‐Hsing

doi:10.1016/j.ins.2005.04.004

Cited by 54 publications

(28 citation statements)

References 27 publications

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“…In the case where both faulty vertices and faulty edges are considered, Huang et al [91] and Chen et al [29], independently, showed that T Q n is (n − 2)-fault-tolerant hamiltonian for any odd integer n 3. This result was improved by Chang et al [23] and Yang et al [162], independently. As regards to fault-tolerant panconnectivity of T Q n , we have known the following results.…”

Section: Twisted Cubesmentioning

confidence: 84%

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Survey on path and cycle embedding in some networks

2009

Front. Math. China

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To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

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Section: Twisted Cubesmentioning

confidence: 84%

“…Chang et al[23] and Yang et al[162]) T Q n is (n − 2)-fault-tolerant pancyclic for any odd integer n 3.…”

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confidence: 99%

Survey on path and cycle embedding in some networks

2009

Front. Math. China

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“…It was proven that a cycle of length l can be embedded with dilation 1 in T Q n − F for any fault-set F ∈ V (T Q n ) ∪ E(T Q n ) with |F | ≤ n − 2 and any integer l with 4 ≤ l ≤ 2 n (n ≥ 3) [19]. In other words, T Q n − F is pancyclic for any fault-set F ∈ V (T Q n ) ∪ E(T Q n ) with |F | ≤ n − 2.…”

Section: Discussionmentioning

confidence: 99%

Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

2007

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In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4 ≤ l ≤ 2 n , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n ≥ 3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4 ≤ l ≤ 2 n , and a given edge (x, y) in an n-dimensional twisted cube, n ≥ 3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x, y) is in C in the twisted cube. Based on the proof of the edgepancyclicity of twisted cubes, we further provide an O(l log l + n 2 + nl) algorithm to find a cycle C of length l that contains (u, v) in T Q n for any (u, v) ∈ E(T Q n ) and any integer l with 4 ≤ l ≤ 2 n .

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“…Embedding paths and cycles in various well-known networks, such as the hypercube and some well-known variations of the hypercube, have been extensively investigated in the literature (see, e.g., Tsai [5] for the hypercubes, Fu [6] for the folded hypercubes, Huang et al [7] and Yang et al [8] for the crossed cubes, Yang et al [9] for the twisted cubes, Hsieh and Chang [10] for the Möbius cubes, Li et al [11] for the star graphs and Xu and Ma [12] for a survey on this topic). Recently, Cao et al [13] have shown that every edge of is contained in cycles of every length from 4 to 2 except 5, and every pair of vertices with distance is connected by paths of every length from to 2 − 1 except 2 and 4 if = 1, from which contains a Hamilton cycle for ⩾ 2 and a Hamilton path between any pair of vertices for ⩾ 3.…”

Section: Introductionmentioning

confidence: 99%