Multiply-twisted hypercube with five or more dimensions is not vertex-transitive

Kulasinghe, P.; Bettayeb, Saïd

doi:10.1016/0020-0190(94)00167-w

Cited by 38 publications

(10 citation statements)

References 4 publications

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“…Twisted N-cube [11] n n Ϫ 1 n No O(n) ? Multiply twisted cube/ n (n ϩ 1)/2 n [15] No [17] O(n 2 ) Yes Crossed cube [10] Flip Mcube [25] n (n ϩ 1)/2 ? ?…”

Section: Various Definitions Of Augmented Cubesmentioning

confidence: 99%

Augmented cubes

2002

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Following the recursive definition of the hypercube Q n , we define the augmented cube AQ n . After showing that its graph is vertex-symmetric, (2n ؊ 1)-regular, and (2n ؊ 1)-connected and that it has diameter  n/2 , we describe optimal routing and broadcasting procedures. The augmented cube possesses several embeddable properties that the hypercube and its variations do not possess.

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“…Twisted N-cube [11] n n Ϫ 1 n No O(n) ? Multiply twisted cube/ n (n ϩ 1)/2 n [15] No [17] O(n 2 ) Yes Crossed cube [10] Flip Mcube [25] n (n ϩ 1)/2 ? ?…”

Section: Various Definitions Of Augmented Cubesmentioning

confidence: 99%

Augmented cubes

2002

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“…Furthermore, CQ n has a diameter of (n + 1)/2t , superior to Q n . Moreover, CQ n is not vertextransitive if n 5 proved by Kulasinghe and Bettayeb [103] and not edgetransitive if n 3 proved by Huang and Xu [92]. This lack of symmetry removes the crossed cubes from the class of Cayley graphs if n 5.…”

Section: Theorem 35 (Fu [56]) F Q N Contains a Fault-free Cycle Withmentioning

confidence: 92%

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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“…[20] proved that the n-dimensional crossed cube CQ n is not vertex-transitive and [21] showed that the vertex set of CQ n can be divided into 2 ⌈(n−4)/2⌉ equivalence classes, where the vertices in one equivalence class are all similar. However, [25] proved that the n-dimensional locally twisted cube LTQ n satisfies the even-oddvertex-transitive property, thus there are only two equivalence classes in LTQ n .…”

Section: Graph Terminology and Notationmentioning

confidence: 99%