Basic tetravalent oriented graphs of independent-cycle type (original) (raw)

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

Journal of Algebraic Combinatorics, 2017

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a subfamily of 'basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

A normal quotient analysis for some families of oriented four-valent graphs

Ars Mathematica Contemporanea, 2017

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.

Four-Valent Oriented Graphs of Biquasiprimitive Type

arXiv: Combinatorics, 2019

Let mathcalOG(4)\mathcal{OG}(4)mathcalOG(4) denote the family of all graph-group pairs (Gamma,G)(\Gamma,G)(Gamma,G) where Gamma\GammaGamma is 4-valent, connected and GGG-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs (Gamma,G)inmathcalOG(4)(\Gamma, G) \in\mathcal{OG}(4)(Gamma,G)inmathcalOG(4) for which every nontrivial normal subgroup of GGG has at most two orbits on the vertices of Gamma\GammaGamma. In particular we show that GGG has a unique minimal normal subgroup NNN and that NcongTkN \cong T^kNcongTk for a simple group TTT and kin1,2,4,8k\in \{1,2,4,8\}kin1,2,4,8. This provides a crucial step towards a general description of the long-studied family mathcalOG(4)\mathcal{OG}(4)mathcalOG(4) in terms of a normal quotient reduction. We also give several methods for constructing pairs (Gamma,G)(\Gamma, G)(Gamma,G) of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup NNN.

Finite edge-transitive oriented graphs of valency four: a global approach

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

Journal of Combinatorial Theory, Series B, 1999

In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not transitive on the arcs of X. Such a graph X is said to be (G, 1Â2)-transitive. The group G induces an orientation of the edges of X, and a certain class of cycles of X (called alternating cycles) determined by the group G is identified as having an important influence on the structure of X. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown that X is a cover of a finite, connected, 4-valent, (G, 1Â2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they are tightly attached, loosely attached, or antipodally attached. We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples.

Biquasiprimitive Oriented Graphs of Valency Four

2017 MATRIX Annals

In this short note we describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex-and edge-transitive group of automorphisms which preserves the edge orientation. In the first article on this topic (Al-bar et al. Electr J Combin 23, 2016), a subfamily of these graphs was identified as 'basic' in the sense that all graphs in this family are normal covers of at least one 'basic' member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types have been analysed in some detail. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. We describe our approach and mention some early results. This work is ongoing. It began at the Tutte Memorial MATRIX Workshop.

Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

Journal of the Australian Mathematical Society, 2002

Finite graphs of valency 4 and girth 4 admitting 1/2-transitive group actions, that is, vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1/2-transitive if its automorphism group acts 1/2-transitively. There is a natural orientation of the edge set of a 1/2-transitive graph induced and preserved by its automorphism group. It is proved that in a finite 1/2-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to 2 .

Classification of tetravalent 2-transitive non-normal Cayley graphs of finite simple groups

2021

A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...

Tetravalent one-regular graphs of order 4p2

Filomat, 2014

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper tetravalent one-regular graphs of order 4p2, where p is a prime, are classified.

A4-Graph of Finite Simple Groups

Iraqi journal of science, 2021

Let G be a finite group and X be a conjugacy class of order 3 in G. In this paper, we introduce a new type of graphs, namely A4-graph of G, as a simple graph denoted by A4(G,X) which has X as a vertex set. Two vertices, x and y, are adjacent if and only if x≠y and x y-1=y x-1. General properties of the A4-graph as well as the structure of A4(G,X) when G@ 3D4(2) will be studied.