s-Regular cyclic coverings of the three-dimensional hypercube Q3

Feng, Yan-Quan; Wang, Kaishun

doi:10.1016/s0195-6698(03)00055-6

Cited by 41 publications

(29 citation statements)

References 26 publications

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“…Since K 4 4 is such a cover (see Lemma 2.9), it follows that X ∼ = K 4 4 for K ∼ = Z 2 . If K ∼ = Z 3 and Z 6 , then it had been proved in [10] that X ∼ = X 1 (3) and X(6) defined in Sect. 1, respectively.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Proof In [10], the coverings of cube whose covering transformation groups are cyclic groups K and whose fiber-preserving automorphism groups act s-regularly are determined. In particular, if s = 2, that is, fiber-preserving automorphism groups act 2-arc-transitively, then there are, respectively, only one 2-fold cover, 3-fold cover and 6-fold cover.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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2-Arc-transitive cyclic covers of K n,n −nK 2

2013

J Algebr Comb

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Du et al. (in J. Comb. Theory B 74:276-290, 1998 and J. Comb. Theory B 93:73-93, 2005), classified regular covers of complete graph whose fiberpreserving automorphism group acts 2-arc-transitively, and whose covering transformation group is either cyclic or isomorphic to Z 2 p or Z 3 p with p a prime. In this paper, a complete classification is achieved of all the regular covers of bipartite complete graphs minus a matching K n,n − nK 2 with cyclic covering transformation groups, whose fiber-preserving automorphism groups act 2-arc-transitively.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

2-Arc-transitive cyclic covers of K n,n −nK 2

2013

J Algebr Comb

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“…Not surprisingly arc-transitive graphs, and cubic arc-transitive graphs in particular, have received considerable attention over the years, the aim being to obtain structural results and possibly a classification of such graphs of different transitivity degrees, particular orders or satisfying additional properties (see, for example [27,30,31,44,45,46,47,60,70,106,107,108,109,110,130]). The frequently used methods in this respect are based on covering graph techniques while using a particular additional condition about their automorphism groups such as, for example, imprimitivity or existence of particular semiregular automorphisms (see Sections 2 and 4).…”

Section: Structural Propertiesmentioning

confidence: 99%

Recent trends and future directions in vertex-transitive graphs

Kutnar

Marušič

2008

Ars Math. Contemp.

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A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group.Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs -the Hamilton path/cycle problem and the semiregularity problem -as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs. In doing so we will also try to contemplate possible directions this area of research is likely to take in the near future. * Supported by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285.E-mail addresses: klavdija.kutnar@upr.si (Klavdija Kutnar), dragan.marusic@upr.si (Dragan Marušič) Copyright c 2008 DMFA -založništvo K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 113The article is organized as follows. In Section 2 we discuss the problem, posed by the second author in 1981 (see [79]) who asked if it is true that a vertex-transitive digraph contains a nonidentity automorphism with all orbits of equal length, in short, a semiregular automorphism. Section 3 gives a quick overview of the problem, posed by Lovàsz in 1969 (see [72]), who asked if it is true that every connected vertex-transitive graph contains a Hamilton path, thus motivating a great deal of research into vertex-transitive graphs in the following decades. In Section 4, imprimitivity, one of the most fundamental concepts in the theory of permutation groups, is considered with special emphasis given to a recently developed theory which links the existence of blocks of imprimitivity in vertex-transitive graphs, having an abelian semiregular subgroup of automorphisms, to certain conditions that need to be satisfied in the corresponding quotient graph relative to the orbits of such a subgroup. Finally, in Section 5 we deal with some recent structural results on arc-transitive and half-arc-transitive graphs, as well as their link to some open problems in the theory of configurations.For group-theoretic terms not defined here we refer the reader to [124]. Semiregularit...

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“…For example, Malnič et al [27] and Feng et al [10] classified connected cubic semisymmetric or s-regular cyclic coverings of the bipartite graph K 3,3 for each 1 ≤ s ≤ 5 when the fibre-preserving group contains an edge-, but not vertex-transitive or an arc-transitive subgroup, respectively. The s-regular cyclic or elementary abelian coverings of the hypercube Q 3 were classified in [11,12] for each 1 ≤ s ≤ 5 when the fibre-preserving group is arc-transitive. Furthermore, using a method developed in [25,26], Malnič and Potočnik [24] classified connected vertex-transitive elementary abelian coverings of the Petersen graph when the fibre-preserving group is vertex-transitive.…”

Section: Main Theoremmentioning

confidence: 99%

Constructing even radius tightly attached half-arc-transitive graphs of valency four

2007

Self Cite

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A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41-76, 1998) classified odd radius tightly attached tetravalent half-arctransitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K 4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.

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s-Regular cyclic coverings of the three-dimensional hypercube Q3

Cited by 41 publications

References 26 publications

2-Arc-transitive cyclic covers of K n,n −nK 2

2-Arc-transitive cyclic covers of K n,n −nK 2

Recent trends and future directions in vertex-transitive graphs

Constructing even radius tightly attached half-arc-transitive graphs of valency four

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