315 research outputs found
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
Vertex-transitive graphs that have no Hamilton decomposition
It is shown that there are infinitely many connected vertex-transitive graphs
that have no Hamilton decomposition, including infinitely many Cayley graphs of
valency 6, and including Cayley graphs of arbitrarily large valency.Comment: This version includes observations that some more of our graphs are
Cayley graphs, and some revisions with new notation. It also includes some
additional concluding remarks, and some updating of reference
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
Vertex-transitive CIS graphs
A CIS graph is a graph in which every maximal stable set and every maximal
clique intersect. A graph is well-covered if all its maximal stable sets are of
the same size, co-well-covered if its complement is well-covered, and
vertex-transitive if, for every pair of vertices, there exists an automorphism
of the graph mapping one to the other. We show that a vertex-transitive graph
is CIS if and only if it is well-covered, co-well-covered, and the product of
its clique and stability numbers equals its order. A graph is irreducible if no
two distinct vertices have the same neighborhood. We classify irreducible
well-covered CIS graphs with clique number at most 3 and vertex-transitive CIS
graphs of valency at most 7, which include an infinite family. We also exhibit
an infinite family of vertex-transitive CIS graphs which are not Cayley
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
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
Normal Edge-Transitive Cayley Graphs of Frobenius Groups
A Cayley Graph for a group is called normal edge-transitive if it admits
an edge-transitive action of some subgroup of the Holomorph of (the
normaliser of a regular copy of in ). We complete
the classification of normal edge-transitive Cayley graphs of order a product
of two primes by dealing with Cayley graphs for Frobenius groups of such
orders. We determine the automorphism groups of these graphs, proving in
particular that there is a unique vertex-primitive example, namely the flag
graph of the Fano plane
Tetravalent edge-transitive Cayley graphs of Frobenius groups
In this paper, we give a characterization for a class of edge-transitive
Cayley graphs, and provide methods for constructing Cayley graphs with certain
symmetry properties. Also this study leads to construct and characterise a new
family of half-transitive graphs
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 , where is a prime,
is given. As a result, there are sporadic and one infinite family of such
graphs, of which the sporadic ones occur when , or , and the
infinite family exists if and only if , 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
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
is a normal graph except for four known graphs. If further, is a
Cayley graph of a non-abelian metacyclic group, then is
half-transitive
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