Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A $0$-type graph can be represented as the graph $\mathrm{BCay}(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{\{h_0,g_1\} : h,g \in H, g h^{-1} \in S\}$. A bi-Cayley graph $\mathrm{BCay}(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\mathrm{BCay}(H,T),$ $\mathrm{BCay}(H,S) \cong \mathrm{BCay}(H,T)$ implies that $T = h S^\alpha$ for some $h \in H$ and $\alpha \in \mathrm{Aut}(H)$. It is also shown that every cubic connected arc-transitive $0$-type bi-Cayley graph over an abelian group is a BCI-graph

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

Heptavalent symmetric graphs with solvable stabilizers admitting vertex-transitive non-abelian simple groups

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group $G$ of automorphisms, then either $G$ is normal in $\rm Aut(\Gamma)$, or $\rm Aut(\Gamma)$ contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is explicitly given as one of $11$ possible exception pairs of non-abelian simple groups. Furthermore, if $G$ is regular on the vertex set of $\Gamma$ then the exception pair $(G,T)$ is one of $7$ possible pairs, and if $G$ is arc-transitive then the exception pair $(G,T)=(A_{17},A_{18})$ or $(A_{35},A_{36})$.Comment: 9. arXiv admin note: substantial text overlap with arXiv:1701.0118

A classification of nilpotent 3-BCI groups

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph \bcay(G,S) is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S) \cong \bcay(G,T) implies that $T = g S^\alpha$ for some $g \in G$ and \alpha \in \aut(G). A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\Z_{2^r},$ $\Z_2^r$ and \Q_8

Arc-transitive Cayley graphs on non-ableian simple groups with soluble vertex stabilizers and valency seven

In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let \Ga be such a Cayley graph on a non-abelian simple group $T$. It is proved that either \Ga is a normal Cayley graph or \Ga is $S$-arc-transitive, with (S,T)=(\A_n,\A_{n-1}) and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on \A_{n-1} and specific examples are constructed

Nowhere-zero 3-flows in graphs admitting solvable arc-transitive groups of automorphisms

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.Comment: This is the final version to be published in: Ars Mathematica Contemporanea (http://amc-journal.eu/index.php/amc

Cubic vertex-transitive non-Cayley graphs of order 12p

A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph X does not tend to infinity as the number of vertices of X tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author. However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when X is either a Cayley graph or an arc-transitive graph.Comment: 18 pages, 1 figur

On cubic symmetric non-Cayley graphs with solvable automorphism groups

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.Comment: 8 page

Arc-transitive bicirculants

In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of seven sporadic arc-transitive graphs. Moreover, each of these ``basic'' graphs is either an arc-transitive bicirculant or an arc-transitive circulant, and each graph in the latter case has an arc-transitive bicirculant normal $r$-cover for some integer $r$