On 2-Arc-Transitive Covers of Complete Graphs

Du, Shihong; Marušič, Dragan; Waller, Adrian

doi:10.1006/jctb.1998.1847

Cited by 68 publications

(66 citation statements)

References 8 publications

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“…Thus, Q 3 × φ Z n is also a regular covering of the complete graph K 4 of order 4. In fact, using results from [9,18,19] one may prove that Q 3 × φ Z n is isomorphic to the following graph G(k, D 2n ). , c), (1, c)), ((0, c), (2, c)), ((0, c), (3, c)), ((1, c), (2, ac)), ((1, c), (3, ab k+1 c)), ((2, c), (3, abc)) | c ∈ D 2n }.…”

Section: Derived Coverings and A Lifting Problemmentioning

confidence: 98%

“…Actually, this new infinite family of cubic 1-regular graphs consists of cyclic-covering graphs of the Hypercube Q 3 , which has a larger degree of symmetry (it is 2-regular), and they are also dihedral-coverings of the complete graph K 4 of order 4 (see Remark later). However, using results from [9,18,19] it may be easily seen that there are no 1-regular cyclic-coverings of Let k be a positive integer greater than 1 and …”

Section: Introductionmentioning

confidence: 99%

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Constructing an Infinite Family of Cubic 1-Regular Graphs

Feng

Kwak

2002

European Journal of Combinatorics

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A graph is 1-regular if its automorphism group acts regularly on the set of its arcs. Miller [J. Comb. Theory, B, 10 (1971), 163-182] constructed an infinite family of cubic 1-regular graphs of order 2 p, where p ≥ 13 is a prime congruent to 1 modulo 3. Marušič and Xu [J. Graph Theory, 25 (1997), 133-138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al. [J. Aust. Math. Soc. A, 56 (1994), 391-402] constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller's construction can be generalized to an infinite family of cubic 1-regular graphs of order 2n, where n ≥ 13 is odd such that 3 divides ϕ(n), the Euler function of n. In this paper, we construct an infinite family of cubic 1-regular graphs with order 8(k 2 + k + 1)(k ≥ 2) as cyclic-coverings of the three-dimensional Hypercube.

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Section: Derived Coverings and A Lifting Problemmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Constructing an Infinite Family of Cubic 1-Regular Graphs

Feng

Kwak

2002

European Journal of Combinatorics

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“…These include: counting isomorphism classes of coverings and, more generally, graph bundles, as considered by Hofmeister [15] and Kwak and Lee [17,18]; constructions of regular maps on surfaces based on covering space techniques due to Archdeacon, Gvozdjak, Nedela, Richter,Širáň,Škoviera and Surowski [1,2,14,28,29,37]; and construction of transitive graphs with a prescribed degree of symmetry, for instance by Du, Malnič, Nedela, Marušič, Scapellato, Seifter, Trofimov and Waller [8,22,23,25,26,34]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf.…”

Section: Historymentioning

confidence: 99%

Lifting Graph Automorphisms by Voltage Assignments

Malnič

Nedela

Škoviera

2000

European Journal of Combinatorics

151

145

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The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

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“…Praeger initiated a systematic study of 2-arc-transitive graphs [27] by showing that all nonbipartite examples are covers of 2-arc-transitive graphs where the automorphism group is quasiprimitive on vertices. This motivated the O'Nan-Scott Theorem for quasiprimitive permutation groups and has led to much work on both quasiprimitive 2-arc-transitive graphs [10,11,15,17,21] and their covers [8,9]. We also refer the reader to the surveys [28,29].…”

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confidence: 99%

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

Giudici

2006

J Algebr Comb

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The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements.

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On 2-Arc-Transitive Covers of Complete Graphs

Cited by 68 publications

References 8 publications

Constructing an Infinite Family of Cubic 1-Regular Graphs

Constructing an Infinite Family of Cubic 1-Regular Graphs

Lifting Graph Automorphisms by Voltage Assignments

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

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