Half-arc-transitive graphs and chiral hypermaps

d'Azevedo, Antonio Breda; Nedela, Roman

doi:10.1016/j.ejc.2003.07.003

Cited by 7 publications

(10 citation statements)

References 16 publications

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“…6 , sz 6 = −z 6 , sz 7 = z 8 − z 9 , sz 8 = z 7 − z 9 and sz 9 = −z 9 . Since (β * 1 ) 2 = 1, one has s 2 = 1 in the ring Z p m .…”

Section: The Proof Of Theorem 21mentioning

confidence: 99%

“…Constructing and characterizing tetravalent 1 2 -arc-transitive graphs is currently an active topic in algebraic graph theory (see [2, 5, 9, 13, 19, 22, 23, 29, 31-33, 35, 37-40, 46, 47]). Apart from research in regular maps related to 1 2 -arc-transitive graphs (see [6] and [32]), and research regarding primitivity/imprimitivity of action for 1 2 -arc-transitive graphs (see [18] and [42]), there are two other directions research in 1 2 -arc-transitive graphs has taken. The first one deals with the study of tetravalent 1 2 -arc-transitive graphs with large order vertex stabilizers, initiated by Marušič and Nedela [34].…”

Section: Introductionmentioning

confidence: 99%

“…Since β * 2 exists, we have 1 β * 2 = t for some t ∈ Z * p m . By Table 2, we have tz 1 = z 2 − z 3 , tz 2 = z 1 − z 3 , tz 3 = −z 3 , tz 4 = z 9 − z 7 , tz 5 = −z 7 , tz 6 = z 8 − z 7 , tz 7 = −z 5 , tz 8 = z 6 − z 5 and tz 9 = z 4 − z 5 . Since (β * 2 ) 2 = 1, one has t 2 = 1.…”

mentioning

confidence: 99%

“…By Table 1, 6 , sz 6 = −z 6 , sz 7 = z 8 − z 9 , sz 8 = z 7 − z 9 and sz 9 = −z 9 . By Table 2, z 4 = z 4 and since 4 = 1, one has 3 = −1 .…”

mentioning

confidence: 99%

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

2007

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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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“…6 , sz 6 = −z 6 , sz 7 = z 8 − z 9 , sz 8 = z 7 − z 9 and sz 9 = −z 9 . Since (β * 1 ) 2 = 1, one has s 2 = 1 in the ring Z p m .…”

Section: The Proof Of Theorem 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…By Table 1, 6 , sz 6 = −z 6 , sz 7 = z 8 − z 9 , sz 8 = z 7 − z 9 and sz 9 = −z 9 . By Table 2, z 4 = z 4 and since 4 = 1, one has 3 = −1 .…”

mentioning

confidence: 99%

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

2007

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“…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%

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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Half-Regular Cayley Maps

Jajcay

Nedela

2014

Graphs and Combinatorics

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Half-arc-transitive graphs and chiral hypermaps

Cited by 7 publications

References 16 publications

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

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

On quartic half-arc-transitive metacirculants

Half-Regular Cayley Maps

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