1,545 research outputs found
Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups
Given a finite group , a regular -balanced Cayley map (RBCM
for short) is a regular Cayley map such that
for all .
In this paper, we clarify a connection between quotients of polynomial rings
and RBCM's on abelian groups, so as to propose a new approach for
classifying RBCM's. We obtain many new results, in particular, a complete
classification for RBCM's on abelian 2-groups.Comment: The previous version of this paper is entitled "A new approach to
regular balanced Cayley maps on abelian -groups
Regular Cayley maps on dihedral groups with the smallest kernel
Let be a regular Cayley map on the dihedral group
of order and let be the power function associated
with . In this paper it is shown that the kernel Ker of the
power function is a dihedral subgroup of and if then the
kernel Ker is of order at least . Moreover, all are
classified for which Ker is of order . In particular, besides
sporadic maps on and vertices respectively, two infinite families
of non--balanced Cayley maps on are obtained
Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps
This paper deals with the Cayley graph
where the generating set consists of all block transpositions. A motivation for
the study of these particular Cayley graphs comes from current research in
Bioinformatics. As the main result, we prove that
Aut is the product of the left translation
group by a dihedral group of order . The proof uses
several properties of the subgraph of
induced by the set . In particular,
is a -regular graph whose automorphism group is
has as many as maximal cliques of size
and its subgraph whose vertices are those in these cliques is a
-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique
cyclic subgroup of of order with regular Cayley maps
on is also discussed. It is shown that the product of the left
translation group by the latter group can be obtained as the automorphism group
of a non--balanced regular Cayley map on
A classification of prime-valent regular Cayley maps on some groups
A Cayley map is a 2-cell embedding of a Cayley graph into an orientable
surface with the same local orientation induced by a cyclic permutation of
generators at each vertex. In this paper, we provide classifications of
prime-valent regular Cayley maps on abelian groups, dihedral groups and
dicyclic groups. Consequently, we show that all prime-valent regular Cayley
maps on dihedral groups are balanced and all prime-valent regular Cayley maps
on abelian groups are either balanced or anti-balanced. Furthermore, we prove
that there is no prime-valent regular Cayley map on any dicyclic group
Classification of reflexibile regular Cayley maps for dihedral groups
In this paper, we classify reflexible regular Cayley maps for dihedral
groups.Comment: 19 page
The Cayley isomorphism property for Cayley maps
In this paper we study finite groups which have Cayley isomorphism property
with respect to Cayley maps, CIM-groups for a brief. We show that the structure
of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a}
where a short list of possible candidates for CIM-groups is given.
Theorem~\ref{111015c} provides concrete examples of infinite series of
CIM-groups
Regular balanced Cayley maps on
A {\it regular balanced Cayley map} (RBCM for short) on a finite group
is an embedding of a Cayley graph on into a surface, with
some special symmetric property. People have classified RBCM's for cyclic,
dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In this
paper we classify RBCM's on the group for each prime number
.Comment: 14 pages, to appear on Discrete Mathematic
Smooth skew-morphisms of the dihedral groups
A skew-morphism of a finite group is a permutation on such
that and for all
where is an integer function. A
skew-morphism is smooth if for all . The
concept of smooth skew-morphisms is a generalization of that of -balanced
skew-morphisms. The aim of the paper is to develop a general theory of smooth
skew-morphisms. As an application we classify smooth skew-morphisms of the
dihedral groups.Comment: 23page
Rotational circulant graphs
A Frobenius group is a transitive permutation group which is not regular but
only the identity element can fix two points. Such a group can be expressed as
the semi-direct product of a nilpotent normal subgroup
and another group fixing a point. A first-kind -Frobenius graph is a
connected Cayley graph on with connection set an -orbit on
that generates , where has an even order or is an involution. It is
known that the first-kind Frobenius graphs admit attractive routing and
gossiping algorithms. A complete rotation in a Cayley graph on a group with
connection set is an automorphism of fixing setwise and permuting
the elements of cyclically. It is known that if the fixed-point set of such
a complete rotation is an independent set and not a vertex-cut, then the
gossiping time of the Cayley graph (under a certain model) attains the smallest
possible value. In this paper we classify all first-kind Frobenius circulant
graphs that admit complete rotations, and describe a means to construct them.
This result can be stated as a necessary and sufficient condition for a
first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable
surface as a balanced regular Cayley map. We construct a family of
non-Frobenius circulants admitting complete rotations such that the
corresponding fixed-point sets are independent and not vertex-cuts. We also
give an infinite family of counterexamples to the conjecture that the
fixed-point set of every complete rotation of a Cayley graph is not a
vertex-cut.Comment: Final versio
Skew product groups for monolithic groups
Skew morphisms, which generalise automorphisms for groups, provide a
fundamental tool for the study of regular Cayley maps and, more generally, for
finite groups with a complementary factorisation , where is cyclic
and core-free in . In this paper, we classify all examples in which is
monolithic (meaning that it has a unique minimal normal subgroup, and that
subgroup is not abelian) and core-free in . As a consequence, we obtain a
classification of all proper skew morphisms of finite non-abelian simple
groups
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