Edge-pancyclicity of recursive circulants

Araki, Tomoyuki

doi:10.1016/j.ipl.2003.09.003

Cited by 26 publications

(29 citation statements)

References 14 publications

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“…Previously, the problem was solved (i.e., constructing cycles of lengths ranging from three to |V (G)| for any e) on recursive circulants [2] and coupled graphs [19]. In addition, cycles of lengths ranging from four to |V (G)| for any e were constructed for crossed cubes [10] and Möbius cubes [26].…”

Section: Discussionmentioning

confidence: 99%

Embedding fault-free cycles in crossed cubes with conditional link faults

Hung

Chen

2008

J Supercomput

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The crossed cube, which is a variation of the hypercube, possesses some properties that are superior to those of the hypercube. In this paper, we show that with the assumption of each node incident with at least two fault-free links, an ndimensional crossed cube with up to 2n − 5 link faults can embed, with dilation one, fault-free cycles of lengths ranging from 4 to 2 n . The assumption is meaningful, for its occurrence probability is very close to 1, and the result is optimal with respect to the number of link faults tolerated. Consequently, it is very probable that algorithms executable on rings of lengths ranging from 4 to 2 n can be applied to an n-dimensional crossed cube with up to 2n − 5 link faults.

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Section: Discussionmentioning

confidence: 99%

Embedding fault-free cycles in crossed cubes with conditional link faults

Hung

Chen

2008

J Supercomput

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“…≤ T (n − 2, u (3) , v (3) , 2 n−2 ) + T (n − 2, u (4) , v (4) where (u , v ) ∈ E(T Q n−2 ). If 2 n−2 + 3 ≤ l ≤ 2 n−1 , T (n, u, v, l) = T n − 2, x (1) , y (1) , l 2 + T n − 2, x (2) , y (2) , (3) , v (3) , 2 n−2 ) + T (n − 2, y (4) , v (4) , 2 n−2 )…”

Section: Beginunclassified

“…If 2 n−2 + 3 ≤ l ≤ 2 n−1 , T (n, u, v, l) = T n − 2, x (1) , y (1) , l 2 + T n − 2, x (2) , y (2) , (3) , v (3) , 2 n−2 ) + T (n − 2, y (4) , v (4) , 2 n−2 )…”

Section: Beginunclassified

“…Based on the proof of the edge-pancyclicity of twisted cubes, we will 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 . The issues on node-pancyclicity and edge-pancyclicity have been discussed in [3,4,6,10,16], and [17].…”

Section: Introductionmentioning

confidence: 99%

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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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“…We will prove that CQ n is a node-pancyclic graph and an edgepancyclic graph when n 2. Node-pancyclicity and edge-pancyclicity of some other graphs have been discussed [1][2][3]13,14].…”

Section: Introductionmentioning

confidence: 99%