Half-transitive graphs of valency 4 with prescribed attachment numbers

Maru?i?, Dragan; Waller, Adrian

doi:10.1002/(sici)1097-0118(200005)34:1<89::aid-jgt8>3.0.co;2-k

Cited by 24 publications

(28 citation statements)

References 14 publications

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“…, d m − 1} for which t = cm and m = ct, claims (i), (iii) and (iv) follow from (20), (21), (24) Table 1). In particular 3 + r 2 = t, and so equation (26) implies that 2 + 2r = t. Note also that (26) and (30) combined together imply that 2(r − 1) = 0. By (27) we thus have 2(r − 1) = n 2 and t = n 2 + 4.…”

Section: Lemma 52mentioning

confidence: 99%

“…The nature of the action of ϕ will contradict half-arctransitivity of X, which thus proves that (AutX) v ∼ = Z 2 , as claimed. Note that (26) and (33) imply that r 2 = 7 u To see that ϕ is in fact a permutation of V (X), we only need to observe that it is injective. Consider the vertices of Table 2, which are of the form u Table 2 for each i = 1, 2, 3, and that the corresponding congruencies modulo 8 of their superscripts are again the eight possibilities 0, 1, .…”

Section: Lemma 52mentioning

confidence: 99%

“…Suppose now, that n 1 > 2 is odd and squarefree. Then (26) implies that r − 1 = n 2 (recall that r = 1). It follows that r 2 = ((r − 1) + 1) 2 = (r − 1) 2 + 2(r − 1) + 1 = 1, which contradicts (30).…”

Section: Lemma 52mentioning

confidence: 99%

“…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%

“…The intersection of two adjacent G-alternating cycles is called a G-attachment set. The attachment of alternating cycles concept has been addressed in a number of papers [18,25,26,27,33] with a particular attention given to the so called G-tightly attached graphs, that is, graphs where two adjacent G-alternating cycles have every other vertex in common. In other words, their Gattachment number coincides with G-radius.…”

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: Lemma 52mentioning

confidence: 99%

Section: Lemma 52mentioning

confidence: 99%

Section: Lemma 52mentioning

confidence: 99%

Section: Introductory and Historic Remarksmentioning

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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Semi‐transitive graphs

Wilson

2003

Journal of Graph Theory

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In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r ) and MPS(k,N;r ). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marus ic and Praeger. This classification shows that nearly all such graphs are spider-graphs. ß

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Four Constructions of Highly Symmetric Tetravalent Graphs

Hill

Wilson

2012

Journal of Graph Theory

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Given a connected, dart‐transitive, cubic graph, constructions of its Hexagonal Capping and its Dart Graph are considered. In each case, the result is a tetravalent graph which inherits symmetry from the original graph and is a covering of the line graph.Similar constructions are then applied to a map (a cellular embedding of a graph in a surface) giving tetravalent coverings of the medial graph. For each construction, conditions on the graph or the map to make the constructed graph dart‐transitive, semisymmetric or 12‐transitive are considered.

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Half-transitive graphs of valency 4 with prescribed attachment numbers

Cited by 24 publications

References 14 publications

On quartic half-arc-transitive metacirculants

On quartic half-arc-transitive metacirculants

Semi‐transitive graphs

Four Constructions of Highly Symmetric Tetravalent Graphs

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