413 research outputs found
Cubic vertex-transitive graphs of girth six
In this paper, a complete classification of finite simple cubic
vertex-transitive graphs of girth is obtained. It is proved that every such
graph, with the exception of the Desargues graph on vertices, is either a
skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of
an arc-transitive triangulation of a closed hyperbolic surface, or the
truncation of a -regular graph with respect to an arc-transitive dihedral
scheme. Cubic vertex-transitive graphs of girth larger than are also
discussed
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 symmetric graphs having an abelian automorphism group with two orbits
Finite connected cubic symmetric graphs of girth 6 have been classified by K.
Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an
abelian automorphism group with two orbits on the vertex set. In this paper all
cubic symmetric graphs with the latter property are determined. In particular,
with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and
GP(24,5), all the obtained graphs are of girth 6.Comment: Keywords: cubic symmetric graph, Haar graph, voltage graph. This
papaer has been withdrawn by the author because it is an outdated versio
Symmetry properties of generalized graph truncations
In the generalized truncation construction, one replaces each vertex of a
-regular graph with a copy of a graph of order . We
investigate the symmetry properties of the graphs constructed in this way,
especially in connection to the symmetry properties of the graphs and
used in the construction. We demonstrate the usefulness of our
results by using them to obtain a classification of cubic vertex-transitive
graphs of girths , , and .Comment: 20 page
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
Structural and computational results on platypus graphs
A platypus graph is a non-hamiltonian graph for which every vertex-deleted
subgraph is traceable. They are closely related to families of graphs
satisfying interesting conditions regarding longest paths and longest cycles,
for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian
graphs.
In this paper, we first investigate cubic platypus graphs, covering all
orders for which such graphs exist: in the general and polyhedral case as well
as for snarks. We then present (not necessarily cubic) platypus graphs of girth
up to 16---whereas no hypohamiltonian graphs of girth greater than 7 are
known---and study their maximum degree, generalising two theorems of Chartrand,
Gould, and Kapoor. Using computational methods, we determine the complete list
of all non-isomorphic platypus graphs for various orders and girths. Finally,
we address two questions raised by the third author in [J. Graph Theory
\textbf{86} (2017) 223--243].Comment: 20 pages; submitted for publicatio
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
Vertex-transitive Haar graphs that are not Cayley graphs
In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised a
question whether there exist vertex-transitive Haar graphs that are not Cayley
graphs. In this note we construct an infinite family of trivalent Haar graphs
that are vertex-transitive but non-Cayley. The smallest example has 40 vertices
and is the well-known Kronecker cover over the dodecahedron graph ,
occurring as the graph in the Foster census of connected symmetric
trivalent graphs.Comment: 9 pages, 2 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
Lines on smooth polarized -surfaces
For each integer , we give a sharp bound on the number of lines
contained in a smooth complex -polarized -surface in
. In the two most interesting cases of sextics in
and octics in , the bounds are and ,
respectively, as conjectured in an earlier paper.Comment: Substantially revised; finer and more complete result
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