30 research outputs found
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
DiscreteZOO: a Fingerprint Database of Discrete Objects
In this paper, we present DiscreteZOO, a project which illustrates some of
the possibilities for computer-supported management of collections of finite
combinatorial (discrete) objects, in particular graphs with a high degree of
symmetry. DiscreteZOO encompasses a data repository, a website and a SageMath
Package
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
Semiregular automorphisms of cubic vertex-transitive graphs
We characterise connected cubic graphs admitting a vertex- transitive group
of automorphisms with an abelian normal subgroup that is not semiregular. We
illustrate the utility of this result by using it to prove that the order of a
semiregular subgroup of maximum order in a vertex-transitive group of
automorphisms of a connected cubic graph grows with the order of the graph.Comment: This paper settles Problem 6.3 in P. Cameron, M. Giudici, G. Jones,
W. Kantor, M. Klin, D. Maru\v{s}i\v{c}, L. A. Nowitz, Transitive permutation
groups without semiregular subgroups, J. London Math. Soc. 66 (2002),
325--33
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
Characterization of Perfect Matching Transitive Graphs
A graph G is perfect matching transitive, shortly PM-transitive, if for any two perfect matchings M and N of G, there is an automorphism f : V(G) ↦ V(G) such that fe(M) = N, where fe(uv) = f(u)f(v). In this paper, the author proposed the definition of PM-transitive, verified PM-transitivity of some symmetric graphs, constructed several families of PM-transitive graphs which are neither vertex-transitive nor edge-transitive, and discussed PM-transitivity of generalized Petersen 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
Danzer's configuration revisited
We revisit the configuration of Danzer DCD(4), a great inspiration for our
work. This configuration of type (35_4) falls into an infinite series of
geometric point-line configurations DCD(n). Each DCD(n) is characterized
combinatorially by having the Kronecker cover over the Odd graph as its
Levi graph. Danzer's configuration is deeply rooted in Pascal's Hexagrammum
Mysticum. Although the combinatorial configuration is highly symmetric, we
conjecture that there are no geometric point-line realizations with 7- or
5-fold rotational symmetry; on the other hand, we found a point-circle
realization having the symmetry group , the dihedral group of order 14
Asymptotic enumeration of vertex-transitive graphs of fixed valency
Let be a group and let be an inverse-closed and identity-free
generating set of . The \emph{Cayley graph} \Cay(G,S) has vertex-set
and two vertices and are adjacent if and only if . Let
be the number of isomorphism classes of -valent Cayley graphs of
order at most . We show that , as
. We also obtain some stronger results in the case .Comment: 15 page
Classification of vertex-transitive cubic partial cubes
Partial cubes are graphs isometrically embeddable into hypercubes. In this
paper it is proved that every cubic, vertex-transitive partial cube is
isomorphic to one of the following graphs: , for some
, the generalized Petersen graph , the cubic permutahedron,
the truncated cuboctahedron, or the truncated icosidodecahedron. This
classification is a generalization of results of Bre\v{s}ar et al.~from 2004 on
cubic mirror graphs, it includes all cubic, distance-regular partial cubes
(Weichsel, 1992), and presents a contribution to the classification of all
cubic partial cubes