Constructing graphs which are ½-transitive

Alspach, Brian; Marušič, Dragan; Nowitz, Lewis A.

doi:10.1017/s1446788700035564

Cited by 91 publications

(176 citation statements)

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“…We note that graphs X o (m, n; r) correspond to the graphs X(r; m, n) introduced in [18]. We also note that the Holt graph, the smallest half-arc-transitive graph (see [1,10,13]), is isomorphic to X o (3, 9; 2). Example 2.2 For each m ≥ 4 even, n ≥ 4 even, r ∈ Z * n , where r m = 1, and t ∈ Z n , where t(r − 1) = 0, let X e (m, n; r, t) be the graph with vertex set V = {u j i | i ∈ Z m , j ∈ Z n } and edges defined by the following adjacencies:…”

Section: Definitions and Examplesmentioning

confidence: 84%

“…Since σ m = ρ t , equation (1) implies that Combining together (27) and ( (26) and the fact that n ≡ 0 (mod 8). This shows that an appropriate k ∈ Z n does exist, as claimed.…”

Section: Lemma 52mentioning

confidence: 99%

“…If the latter occurs then there exists an automorphism of C p (X) mapping some 3-path with code DA + to some 3-path with code A − A + . It can be seen that any 3-path with code DA + lies on some 8-cycle fromH 4 , whereas the only 8-cycles of C p (X) which contain 3-paths with code A − A + are the 8-cycles fromH 1 . Therefore, some automorphism of C p (X), mapping an 8-cycle fromH 4 to an 8-cycle fromH 1 exists.…”

Section: Lemma 52mentioning

confidence: 99%

“…Interest in the study of this class of graphs reemerged about a decade later following a series of papers dealing mainly with classification of certain restricted classes of such graphs as well as with various methods of constructions of new families of such graphs [1,3,28,29,31,34]; but see also the survey article [19] which covers the respective literature prior to 1998. With some of the research emphasis shifting to questions concerning structural properties of half-arc-transitive graphs, these graphs have remained an active topic of research to this day; see [7,8,11,12,14,15,16,18,20,21,22,23,24,25,26,27,33,35].…”

Section: Introductory and Historic Remarksmentioning

confidence: 99%

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On quartic half-arc-transitive metacirculants

Marušič

Šparl

2007

J Algebr Comb

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Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m, n)-semiregular for some integers m ≥ 1, n ≥ 2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ m has at least one fixed vertex. A half-arc-transitive graph is a vertex-and edge-but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic halfarc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.Keywords: Graph; Metacirculant graph; Half-arc-transitive; Tightly attached; Automorphism group 1 Introductory and historic remarks Throughout this paper graphs are assumed to be finite and, unless stated otherwise, simple, connected and undirected (but with an implicit orientation of the edges when appropriate). For group-theoretic concepts not defined here we refer the reader to [4,9,32], and for graph-theoretic terms not defined here we refer the reader to [5].Given a graph X we let V (X), E(X), A(X) and AutX be the vertex set, the edge set, the arc set and the automorphism group of X, respectively. A graph X is said to be vertex-transitive, edge-transitive and arc-transitive if its automorphism group AutX acts transitively on V (X), E(X) and A(X), respectively. We say that X is half-arctransitive provided it is vertex-and edge-but not arc-transitive. More generally, by a half arc-transitive action of a subgroup G ≤ AutX on X we mean a vertex-and edge-but not arc-transitive action of G on X. In this case we say that the graph X is (G, 1 2 )-arc-transitive, and we say that the graph X is (G, 1 2 , H)-arc-transitive when it needs to be stressed that the vertex stabilizers G v (for v ∈ V (X)) are isomorphic to a particular subgroup H ≤ G. By a classical result of Tutte [30, 7.35, p.59], a graph admitting a half-arc-transitive group action is necessarily of even valency. A 1 Supported in part by "ARRS -Agencija za znanost Republike Slovenije", program no. P1-0285.

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Section: Definitions and Examplesmentioning

confidence: 84%

“…Since σ m = ρ t , equation (1) implies that Combining together (27) and ( (26) and the fact that n ≡ 0 (mod 8). This shows that an appropriate k ∈ Z n does exist, as claimed.…”

Section: Lemma 52mentioning

confidence: 99%

Section: Lemma 52mentioning

confidence: 99%

Section: Introductory and Historic Remarksmentioning

confidence: 99%

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On quartic half-arc-transitive metacirculants

Marušič

Šparl

2007

J Algebr Comb

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“…In 1966 Tutte [121] asked if for every given even integer k ≥ 4 there exists a half-arc-transitive graph of valency k. A positive answer was given by Bouwer in [21] by a construction of a half-arc-transitive graph of valency k for any even integer k ≥ 4. The smallest known example of a half-arc-transitive graph is the Doyle-Holt graph [7,59] which is quartic and of order 27. Graphs admitting a halfarc-transitive group actions are in a one-to-one correspondence with the so called orbital graphs of groups with non-self-paired orbitals whereas graphs admitting arc-transitive group actions are in a one-to-one correspondence with orbital graphs of groups with self-paired K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 119 orbitals.…”

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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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