Sucharita Biswas, Angsuman Das
In this paper, we introduce a new family of graphs, $Γ(n,a)$. We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of $Γ(n,a)$.
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In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant $Γ$ is a normal graph except for four known graphs. If further, $Γ$ is a Cayley graph of a non-abelian metacyclic group, then $Γ$ is half-transitive.
A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order $12p$.
Let $s$ be a positive integer. A graph is $s$-transitive if its automorphism group is transitive on s-arcs but not on $(s + 1)$-arcs. In this paper, we study all tetravalent s-transitive graphs of order $6p^2$.
It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason.
On tetravalent half-arc-transitive graphs
本文完全分类了连通四价半弧传递图的顶点稳定子，证明其恰为非平凡同心群，并首次构造出双拟本原型基本四价半弧传递图的无限族。
Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s\geqslant3$ has been wide ope…
Tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Marušič 20 years ago. In this paper a new parameter for such graphs, giving a further insight into their structure, is introduced. Various properties of this parameter are given and the parameter is completely determined for th…
On tetravalent half-arc-transitive graphs of girth 5
本文研究围长为5的四度半弧传递图，证明除两个例外外其5-圈均为有向且一致，并给出无限族及紧附着图与弱亚循环图的分类。
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 […
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
This paper is to accompany the Census of Edge-Transitive Tetravalent Graphs, available at jan.ucc.nau.edu/~swilson/C4FullSite/index.html, which is a collection of all known edge-transitive graphs of valence 4 up to 512 vertices. The Census contains information for each graph. This information includes parameters such as group order, diameter, girth etc., all known constructions, relations to other graphs in the Census and intersting substructures such as colorings, cycle structures, and dissections. We try to present most graphs as members of one or more parameterized families, and one purpose…
A graph $Γ$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(Γ)$ is transitive on the set of vertices of $Γ$ and the set of $s$-arcs of $Γ$, where for an integer $s \ge 1$ an $s$-arc of $Γ$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $Γ$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $Γ$ is called 2-transitive if it is $(\mathrm{Aut}(Γ), 2)$-arc-transitive but not $(\mathrm{Aut}(Γ), 3)$-arc-transitive. A Cayley graph $Γ$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(Γ)$ and non-normal othe…
In order to complete (and generalize) results of Gardiner and Praeger on 4-valent symmetric graphs (European J. Combin, 15 (1994)) we apply the method of lifting automorphisms in the context of elementary-abelian covering projections. In particular, the vertex- and edge-transitive graphs whose quotient by a normal $p$-elementary abelian group of automorphisms, for $p$ an odd prime, is a cycle, are described in terms of cyclic and negacyclic codes. Specifically, the symmetry properties of such graphs are derived from certain properties of the generating polynomials of cyclic and negacyclic code…
On the radius and the attachment number of tetravalent half-arc-transitive graphs
研究四价半弧传递图半径 r 与附着数 a 的关系，证明当 a 为奇数两倍时 a 不整除 r 的图必为弧传递，并完全刻画 r=3、a=2 的图类。
On automorphism groups of half-arc-transitive tetravalent graphs
本文刻画了存在子群M<H使得图是M-半弧传递且H-弧传递的连通四价图，构造了各类例子，包括对Rivera和Šparl 2019年问题的反例以及首个非交换单群上的四价正规边传递非正规Cayley图。
A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.
Generalized Gardiner-Praeger graphs and their symmetries
本文研究CPM图（Gardiner-Praeger图的推广），确定其自同构群、同构关系，并区分2-弧传递、弧传递非2-弧传递及半弧传递的CPM图类型。
A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Potočnik.
Symmetries of the Woolly Hat graphs
本文研究允许含三个轨道的半正则自同构的四价边传递图（即Woolly Hat图），证明不存在边传递的该类图，并完成了顶点传递的Woolly Hat图的分类。
In this paper, we introduce a family of tetravalent graphs called propeller graphs, denoted by $Pr_{n}(b,c,d)$. We then produce three infinite subfamilies and one finite subfamily of arc-transitive propeller graphs, and show that all such graphs are necessarily members of one of these four subfamilies, up to isomorphism. We close the paper with questions for further investigation, as well as a few conjectures.
Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.
Let $Γ$ be a finite X-symmetric graph with a nontrivial X-invariant partition $\mathcal {B}$ on $V(Γ)$ such that $Γ_{\mathcal {B}}$ is a connected (X,2)-arc-transitive graph and $Γ$ is not a multicover of $Γ_{\mathcal {B}}$. This article aims to give a characterization of $(Γ, X, \mathcal {B})$ for the case where $|Γ(C) \cap B| = 3$ for $B\in \mathcal {B}$ and $C \in Γ_{\mathcal {B}}(B)$. This investigation requires a study on (X,2)-arc-transitive graphs of valency 4 or 7. We give a characterization of tetravalent (X,2)-arc-transitive graphs at first; and as a byproduct, we prove that every te…
Basic Tetravalent Oriented Graphs of Independent-Cycle Type
本文对具有至少两个独立循环正规商的基本四价图-群对进行了显式分类，这些对构成所有四价图-群对的覆盖基础。
Basic Tetravalent Oriented Graphs with Cyclic Normal Quotients
本文刻画了 $\mathcal{OG}(4)$ 中至少有一个循环图作为 $G$-正规商的基本四价定向图，构造了大量新的无限族并解决了若干公开问题，从而完成了对所有基本对的分类。
A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$…
Resolution of a conjecture about linking ring structures
本文使用群合并的方法，解决了P. Potočnik和S. Wilson在研究围长为4的四价半对称图工具LR-结构的论文中遗留的关于连接环结构的猜想及问题。
A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.
Following Alspach and Parsons, a {\em 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 $m \geq 1$, $n \geq 2$, 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 {\em 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 explo…
A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as $B(k,m,n)$, defined for every triple $(k,m,n)$ of integers greater than $1$ with $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9…
Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $λ(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $λ(G)=δ(G)$, and $G$ super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edge-connectivity $λ'(G)$ of $G$ is the minimum number of edges whose removal disconnects $G$ into non-trivial components. If $λ'(G)$ achieves the upper bound of restricted edge-connectivity, then $G$ is said to be $λ'$-optimal. A bipartite multigraph is said to be half-t…
A graph is said to be {\em half-arc-transitive} if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called {\em alternating cycles} is associated, all of which have the same even length. Half of this length is called the {\em radius} of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so called {\em attachment number}, coincides with the radius, we say that the graph is {\em tightly …
A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, ha…
Determining the vertex stabilizers of 4-valent half-arc-transitive graphs
本文建立判定同心群能否作为连通四度半弧传递图顶点稳定子群的通用理论框架，显著扩充已知稳定子群，并证明Spiga–Xia猜想中的ℋ₇×C₂^(m−7)。
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $ρ= 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
On certain edge-transitive bicirculants
本文研究高价边传递双循环图的存在性，构造并分类价6、围长3的例子，得到无限多个弧传递与半弧传递图，并推出具有普遍可达关系的价6半弧传递图无限族。
In this paper we introduce and study a type of Cayley graph -- subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $K_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.
A graph $\G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over $H$. Such a graph $\G$ is called {\em normal\/} if $H$ is normal in the full automorphism group of $\G$, and {\em normal edge-transitive\/} if the normaliser of $H$ in the full automorphism group of $\G$ is transitive on the edges of $\G$. % In this paper, we give a characterisation of normal edge-transitive bi-Cayley graphs, %which form an important subfamily of bi-Cayley graphs, and in particular, we give a detailed description of $2$-arc-…
A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs
本文证明了粗版本的Halin网格定理：每个一端、局部有限且包含无穷多条射线的不交并作为渐近子式的图，也包含半网格作为渐近子式，并应用于拟传递局部有限图，解决了Georgakopoulos和Papasoglu的猜想与问题。
Asymptotic half-grid and full-grid minors
该研究证明两类满足粗端、圈空间由有界长圈生成条件的图分别包含全网格、半网格作为渐近与发散子式，部分解决了相关学者提出的问题。
We study the problem of half-domination sets of vertices in vertex transitive infinite graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp and in the rest, we show upper bounds for the average densities of vertices in half-domination sets.
In this paper, all graphs are assumed to be finite. For $s\geq 1$ and a graph $\G$, if for every pair of isomorphic connected induced subgraphs on at most $s$ vertices there exists an automorphism of $\G$ mapping the first to the second, then we say that $\G$ is $s$-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most $s$ vertices can be extended to an automorphism of $\G$, then we say that $\G$ is $s$-connected-homogeneous. For $n\geq 1$, a graph $\G$ is said to be locally $2\K_n$ if the subgraph $[\G(u)]$ induced on the set of vert…
A graph $Γ$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $Γ$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $Γ$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(Γ)$ of $Γ$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(Γ)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no …
Like the Coxeter graph became reattached into the Klein graph in [2], the Levi graphs of the $9_3$ and $10_3$ self-dual configurations, known as the Pappus and Desargues ($k$-transitive) graphs $\mathcal P$ and $\mathcal D$ (where $k=3$), also admit reattachments of the distance-$(k-1)$ graphs of half of their oriented shortest cycles via orientation assignments on their common $(k-1)$-arcs, concurrent for ${\mathcal P}$ and opposite for $\mathcal D$, now into 2 disjoint copies of their corresponding Menger graphs. Here, $\mathcal P$ is the unique cubic distance-transitive (or CDT) graph with …
A directed graph $G=(V,E)$ is {\it strongly pseudo transitive} if there is a partition $\{A,E-A\}$ of $E$ so that graphs $G_1=(V,A)$ and $G_2=(V,E-A)$ are transitive, and additionally, if $ab\in A$ and $bc\in E $ implies that $ac\in E$. A strongly pseudo transitive graph $G=(V,E)$ is strongly pseudo transitive of the first type, if $ab\in A$ and $bc\in E$ implies $ac\in A$. An undirected graph is co-strongly pseudo transitive (co-strongly pseudo transitive of the first type) if its complement has an orientation which is strongly pseudo transitive (co-strongly pseudo transitive of the first typ…
Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(Γ,G)$ where $Γ$ is 4-valent, connected and $G$-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 $(Γ, G) \in\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $Γ$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \cong T^k$ for a simple group $T$ and $k\in \{1,2,4,8\}$. This provides a crucial step tow…
Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner a…
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