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.Comment: 19 page

Groups of order at most 6 000 generated by two elements, one of which is an involution, and related structures

A (2,*)-group is a group that can be generated by two elements, one of which is an involution. We describe the method we have used to produce a census of all (2,*)-groups of order at most 6 000. Various well-known combinatorial structures are closely related to (2,*)-groups and we also obtain censuses of these as a corollary.Comment: 3 figure

A Classification of Tightly Attached Half-Arc-Transitive Graphs of Valency 4

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 attached}. In {\em J. Combin. Theory Ser. B} {73} (1998) 41--76, Maru\v{s}i\v{c} gave a classification of tightly attached \hatr graphs of valency 4 with odd radius. In this paper the even radius tightly attached graphs of valency 4 are classified, thus completing the classification of all tightly attached half-arc-transitive graphs of valency 4.Comment: 36 pages, 1 figur

On quartic half-arc-transitive metacirculants

Following Alspach and Parsons, a {\em metacirculant graph} is a graph admitting a transitive group generated by two automorphisms $\rho$ and $\sigma$, where $\rho$ is $(m,n)$-semiregular for some integers $m \geq 1$, $n \geq 2$, and where $\sigma$ normalizes $\rho$, cyclically permuting the orbits of $\rho$ in such a way that $\sigma^m$ 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 explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.Comment: 31 pages, 2 figure

Half-arc-transitive graphs of prime-cube order of small valencies

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 $p$-groups is constructed.Comment: 13 page

Sharply kk-arc-transitive-digraphs: finite and infinite examples

A general method for constructing sharply $k$-arc-transitive digraphs, i.e. digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is presented. Using our method it is possible to construct both finite and infinite examples. The infinite examples can have one, two or infinitely many ends. Among the one-ended examples there are also digraphs that have polynomial growth

Cubic vertex-transitive graphs on up to 1280 vertices

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

Lifting a prescribed group of automorphisms of graphs

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of \Aut(P) lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup $G$ of \Aut(P), to find $\wp$ along which $G$ lifts but no further automorphism of $P$ does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group $A_5$ but not the whole symmetric group $S_5$. (Recall that \Aut(P)\cong S_5.) Some other time it is important, given a subgroup $G$ of \Aut(P), to find $\wp$ with the property that \Aut(\tilde{P}) is the lift of $G$. Typically, it is desirable to find $\wp$ satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is $S_5$, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just $A_5$. In this paper, we address this problem in full generality.Comment: 10 page

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism

Derangement action digraphs and graphs

We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of \Der(X), the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on $S$ under which $\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given.Comment: 15 pages, 1 figur