162 research outputs found
5-Arc transitive cubic Cayley graphs on finite simple groups
AbstractIn this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group A47; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two non-normal connected cubic arc-transitive Cayley graphs for finite nonabelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs
Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs
A finite simple graph is called a bi-Cayley graph over a group if it has
a semiregular automorphism group, isomorphic to 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 -type when it is bipartite, and the bipartition classes are equal to
the two orbits of the respective semiregular automorphism group. A -type
graph can be represented as the graph where is a
subset of the vertex set of which consists of two copies of say
and and the edge set is . A
bi-Cayley graph is called a BCI-graph if for any bi-Cayley
graph
implies that for some and . It is also shown that every cubic connected arc-transitive
-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
A classification of nilpotent 3-BCI groups
Given a finite group and a subset the bi-Cayley graph
\bcay(G,S) is the graph whose vertex set is and edge set
is . 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 for some and \alpha
\in \aut(G). A group is called an -BCI-group if all bi-Cayley graphs of
of valency at most 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
where is a homocyclic group of odd order, and is trivial or one of the
groups and \Q_8
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
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
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
-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 -cover for some integer
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 -regular graph of type , that is, a graph with no
automorphism of order interchanging two adjacent vertices. In this paper an
infinite family of non-Cayley cubic -regular graphs of type with a
solvable automorphism group is constructed. The smallest graph in this family
has order 6174.Comment: 8 page
Hamiltonicity of Cubic Cayley Graphs
Following a problem posed by Lov\'asz in 1969, it is believed that every
connected vertex-transitive graph has a Hamilton path. This is shown here to be
true for cubic Cayley graphs arising from groups having a
-presentation, that is, for groups G=\la a,b| a^2=1, b^s=1, (ab)^3=1,
etc. \ra generated by an involution and an element of order
such that their product has order 3. More precisely, it is shown that the
Cayley graph has a Hamilton cycle when (and
thus ) is congruent to 2 modulo 4, and has a long cycle missing only two
vertices (and thus necessarily a Hamilton path) when is congruent to 0
modulo 4.Comment: 13 pages, 6 figure
Edge-transitive bi-Cayley graphs
A graph \G admitting a group of automorphisms acting semi-regularly on
the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over
. Such a graph \G is called {\em normal\/} if is normal in the full
automorphism group of \G, and {\em normal edge-transitive\/} if the
normaliser of 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
-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 -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 , and answer some open questions
from the literature about -arc-transitive, half-arc-transitive and
semisymmetric graphs
- …