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

Edge-transitive bi-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-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic $p$-groups. We find that under certain conditions, `normal edge-transitive' is the same as `normal' for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most $6$, and answer some open questions from the literature about $2$-arc-transitive, half-arc-transitive and semisymmetric graphs

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

Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ 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 ${\rm Aut}(\Gamma)$ of $\Gamma$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(\Gamma)$. 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 arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $p$.Comment: 20 pages, 1 figur

A classification of tetravalent edge-transitive metacirculants of odd order

In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant $\Gamma$ is a normal graph except for four known graphs. If further, $\Gamma$ is a Cayley graph of a non-abelian metacyclic group, then $\Gamma$ is half-transitive

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 $r$

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

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

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