A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

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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“…(Note that up to conjugacy in Aut(T 4 ), the above group is the unique f ( a, b, c ), f (g)) (see [20,Section 3] for more information).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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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“…In the rest of the proof, we assume

P = P_{2} (p)

. If

p = 3

, then by , up to isomorphism, there is only one tetravalent HAT graph of order

54

, and by Theorem , we have

normalΓ ≅ Γ_{2, 1, 1, 1}^{1}

and

P ≅ P_{1} (p)

, a contradiction. Thus,

p > 3

.…”

Section: Tetravalent Hat Graphs Of Order 2p3mentioning

confidence: 97%

“…Note that in his results, Bouwer provided just one graph for each even valency k 2 > 2, and this was recently improved by Conder and Žitnik in [7] and by Rivera and Šparl in [20] by showing that infinitely many HAT graphs of any given even valency greater than 2 exist. Following Bouwer's work, HAT graphs have received a great deal of attention, and constructing and characterizing tetravalent HAT graphs, especially in the 4-valent case, is also currently an active topic in algebraic graph theory (see, for instance, [1,[5][6][7]14,15,[17][18][19]21,22,27]).…”

Section: Introductionmentioning

confidence: 99%

Tetravalent half‐arc‐transitive bi‐p‐metacirculants

Zhang

Zhou

2018

Journal of Graph Theory

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A graph is half‐arc‐transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A graph is said to be a bi‐Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly and with two orbits on the vertex set. In this paper, a classification is given of tetravalent half‐arc‐transitive bi‐Cayley graphs over metacyclic p‐groups for each odd prime p, and this is then used to give a classification of tetravalent half‐arc‐transitive graphs of order twice a prime cube.

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“…It should be pointed out that the G-half-arc transitive case has received much attention in the past (see, for example [4,6,8,9,12,15]), so it is only natural to initiate a similar systematic study of the other…”

Section: Linking Rings Structures and Tetravalent Vertex-transitive Gmentioning

confidence: 99%

“…Encouraged by the successful compilation of complete lists of all cubic vertex-transitive graphs of order at most 1280 [11], all tetravalent arc-transitive graphs of order at most 640 and of all tetravalent half-arc-transitive graphs of order at most 1000 (see [12]), one feels tempted to attempt a construction of the list of all ''small' tetravalent vertex-transitive graphs. As will be briefly discussed in Section 1.2, the graphs admitting linking rings structures form an important family of tetravalent vertex-transitive graphs, which will have to be studied in detail if one wants to succeed in this endeavor.…”

Section: Introductionmentioning

confidence: 99%

Linking rings structures and semisymmetric graphs: Cayley constructions

Potočnik

Wilson

2016

European Journal of Combinatorics

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a b s t r a c tAn LR structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR structures were introduced in Potočnik and Wilson (2014) as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we consider algebraic methods of constructing LR structures, using number theory, Cayley graphs, affine groups, abelian groups and fields.

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A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

Cited by 51 publications

References 31 publications

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

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

Tetravalent half‐arc‐transitive bi‐p‐metacirculants

Linking rings structures and semisymmetric graphs: Cayley constructions

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