Automorphisms of augmented cubes

Choudum, S. A.; Sunitha, V.

doi:10.1080/00207160701543384

Cited by 14 publications

(22 citation statements)

References 5 publications

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“…Obviously, AQ n is a (2n − 1)-regular graph with 2 n vertices. It has been shown by Choudum and Sunitha [36][37][38] that AQ n is vertex-symmetric, (2n− 1)-connected for n = 3 (AQ 3 is 4-connected) and has diameter n/2 , the wide-diameter and fault-diameter n/2 + 1 (n 5). At the same time, they showed that AQ n is pancyclic for n 2.…”

Section: Augmented Cubesmentioning

confidence: 98%

“…The n-dimensional augmented cube AQ n (n 1), proposed by Choudum and Sunitha [36][37][38], can be defined recursively as follows: AQ 1 is a complete graph K 2 with the vertex set {0, 1}. For n 2, AQ n is obtained by taking two copies of the augmented cube AQ n−1 , denoted by AQ 0 n−1 and AQ 1 n−1 , and adding 2 × 2 n−1 edges between the two as follows.…”

Section: Augmented Cubesmentioning

confidence: 99%

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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: Augmented Cubesmentioning

confidence: 98%

Section: Augmented Cubesmentioning

confidence: 99%

Survey on path and cycle embedding in some networks

2009

Front. Math. China

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“…However, we will show later, AQ n is neither edge-transitive nor arc-transitive for n ≥ 3. In [13], Choudum and Sunitha find all automorphisms of AQ n . We combine a couple of their results to bound the determining number.…”

Section: Augmented Hypercubesmentioning

confidence: 99%

“…Lemma 3.2 in [13] asserts that if x 1 is the vertex of AQ n that differs from x only in the first position and x n is the vertex that differs only in the n th position, then any automorphism of AQ n that fixes x, x 1 and x n must fix every vertex in N[x]. Further, Theorem 3.3 in [13] asserts that if ϕ, ψ ∈ Aut(AQ n ) satisfy ϕ(a) = ψ(a) for all a ∈ N[x], then ϕ = ψ on AQ n . By definition, this means that every closed neighborhood is a determining set.…”

Section: Augmented Hypercubesmentioning

confidence: 99%

Symmetry Parameters of Various Hypercube Families

Boutin¹,

Cockburn²,

Keough³

et al. 2021

Preprint

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“…Although our primary motivation came from the game of Tic-Tac-Toe, we believe our result has much broader interest as it presents an analogy of automorphism characterization results of hypercubes (see e.g. [7,10]).…”

Section: Motivationmentioning

confidence: 99%

Automorphisms of the Cube $$n^d$$

Dvořák

Valla

2016

Lecture Notes in Computer Science

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Consider a hypergraph H d n where the vertices are points of the d-dimensional combinatorial cube n d and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of H d n , i.e., permutations of points of n d preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of H d n there exists an automorphism of H d n preserving the colors. We show that this problem is GI-complete.

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Automorphisms of augmented cubes

Cited by 14 publications

References 5 publications

Survey on path and cycle embedding in some networks

Survey on path and cycle embedding in some networks

Symmetry Parameters of Various Hypercube Families

Automorphisms of the Cube $$n^d$$

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