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

Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups

Given a finite group $\Gamma$, a regular $t$-balanced Cayley map (RBCM$_{t}$ for short) is a regular Cayley map $\mathcal{CM}(G,\Omega,\rho)$ such that $\rho(\omega)^{-1}=\rho^{t}(\omega)$ for all $\omega\in\Omega$. In this paper, we clarify a connection between quotients of polynomial rings and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for classifying RBCM$_{t}$'s. We obtain many new results, in particular, a complete classification for RBCM$_{t}$'s on abelian 2-groups.Comment: The previous version of this paper is entitled "A new approach to regular balanced Cayley maps on abelian $p$-groups

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

Smooth skew-morphisms of the dihedral groups

A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. A skew-morphism is smooth if $\pi(\varphi(x))=\pi(x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-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

Regular Cayley maps on dihedral groups with the smallest kernel

Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n, n \ge 2,$ and let $\pi$ be the power function associated with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(\pi)$ of the power function $\pi$ is a dihedral subgroup of $D_n$ and if $n \ne 3,$ then the kernel Ker$(\pi)$ is of order at least $4$. Moreover, all $\mathcal{M}$ are classified for which Ker$(\pi)$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained

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 $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups

Complete regular dessins and skew-morphisms of cyclic groups

A dessin is a 2-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph $K_{m,n}$, called $(m,n)$-complete regular dessins. The purpose is to establish a rather surprising correspondence between $(m,n)$-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group $A$ is a bijection $\varphi\colon A\to A$ that satisfies the identity $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for some function $\pi\colon A\to\mathbb{Z}$ and fixes the neutral element of~$A$. We show that every $(m,n)$-complete regular dessin $\mathcal{D}$ determines a pair of reciprocal skew-morphisms of the cyclic groups $\mathbb{Z}_n$ and $\mathbb{Z}_m$. Conversely, $\mathcal{D}$ can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers $m$ and $n$ for which there exists, up to interchange of colours, exactly one $(m,n)$-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order $m$ and $n$ is abelian, which eventually comes down to the condition $\gcd(m,\phi(n))=\gcd(\phi(m),n)=1$, where $\phi$ is Euler's totient function.Comment: 19papge

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 $G = K \rtimes H$ of a nilpotent normal subgroup $K$ and another group $H$ fixing a point. A first-kind $G$-Frobenius graph is a connected Cayley graph on $K$ with connection set an $H$-orbit $a^H$ on $K$ that generates $K$, where $H$ has an even order or $a$ 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 $G$ with connection set $S$ is an automorphism of $G$ fixing $S$ setwise and permuting the elements of $S$ 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

Regular balanced Cayley maps on PSL(2,p){\rm PSL}(2,p)

A {\it regular balanced Cayley map} (RBCM for short) on a finite group $\Gamma$ is an embedding of a Cayley graph on $\Gamma$ 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 ${\rm PSL}(2,p)$ for each prime number $p>3$.Comment: 14 pages, to appear on Discrete Mathematic