Elementary Abelian Covers of Graphs

We present some new discoveries about graphs that are half-arc-transitive (that is, vertex-and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest * The first author was supported by a James Cook Fellowship and a Marsden Fund grant (UOA1015) from the Royal Society of New Zealand with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser D 4 (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.

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“…(For background information on quotients and covering projections of graphs, see [12,20] for example. )…”

Section: Stabiliser Of Ordermentioning

confidence: 99%

Some recent discoveries about half-arc-transitive graphs

2014

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“…As in, say, [12], for us, a graph will be an ordered 4-tuple (D, V ; beg, inv) where D and V = ∅ are disjoint finite sets of darts and vertices, respectively, beg : D → V is a mapping which assigns to each dart x its initial vertex beg x, and inv : D → D is an involution which interchanges every dart x with its inverse dart, also denoted by x −1 . If Γ is a graph, then we let D(Γ) and V(Γ) denote its dart-set and its vertex-set, respectively.…”

Section: Concerning Graphsmentioning

confidence: 97%

“…This section summarises some of the facts about quotients and covers that can be derived easily from [11] or [12].…”

Section: 2mentioning

confidence: 99%

“…Such an algorithm has been developed and implemented in Magma by the second author; it is described in detail in [23], and is based on the theory of elementary abelian regular covering projections, presented in [12]. The essence of this algorithm relies on the fact that for every solvable G-admissible regular covering projection ℘ : Γ → Λ there exists a sequence of elementary abelian regular covering projections ℘ i : Γ i → Γ i−1 for i ∈ {1, .…”

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confidence: 99%

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Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8

Potočnik

Požar

2017

Journal of Combinatorial Theory, Series A

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Abstract. A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in [Conder and Marušič, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Combin. Theory Ser. B 88 (2003) 67-76]. This example has 10752 vertices and vertex-stabiliser isomorphic to the dihedral group of order 8. In this paper, we show that no such graphs of smaller order exist, thus answering a frequently asked question. IntroductionLet Γ be a connected finite graph and G a group of automorphisms of Γ. If G acts transitively on the set of vertices, edges or arcs of the graph (an arc is an ordered pair of adjacent vertices), then Γ is said to be G-vertex-transitive, G-edgetransitive or G-arc-transitive, respectively. Furthermore, if G acts transitively on the vertices, edges, but not arcs of the graph, then Γ is (G, 2 -arc-transitive graphs (and half-arc-transitive graphs in general) were much studied by many authors from different points of view, ranging from purely combinatorial [7,13,17,25,26,27,29], geometrical [15], to permutation group theoretical [1,8,10] and abstract group theoretical [16,21,24].2000 Mathematics Subject Classification. 20B25.

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“…(For graph-covering terms not defined here we refer the reader to [55,102,105].) As shown by Djoković [35], symmetry properties of X are, to some extent, reflected by the symmetries of Y provided that enough automorphisms lift along p. The lifting problem is well understood, see [35,40,44,73,76,77,78]. Thus, studying symmetries of X arising via lifting automorphisms should be considered 'easy'.…”

Section: Imprimitivitymentioning

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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Elementary Abelian Covers of Graphs

Cited by 101 publications

References 21 publications

Some recent discoveries about half-arc-transitive graphs

Some recent discoveries about half-arc-transitive graphs

Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8

Recent trends and future directions in vertex-transitive graphs

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