Vertex-transitive covers of semi-equivelar toroidal maps

Some Semi - Equivelar Maps

Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than $2-$Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics -8 and -10 and on non-orientable surface of Eule…

Equivelar and d-Covered Triangulations of Surfaces. II. Cyclic Triangulations and Tessellations

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cyclic tessellations of surfaces are derived from these triangulated series.

Semi-equivelar maps of Euler characteristics -2 with few vertices

We enumerate and classify all the semi equivelar maps on the surface of $ χ=-2 $ with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.

On enumeration of a class of toroidal graphs

We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We know the classification of the maps of types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$ on the torus. In this article, we attempt to classify maps of types $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $…

半等顶环面映射及其k-边覆盖

Semi-equivelar toroidal maps and their k-edge covers

本文研究了半等顶环面地图的边轨道界限，证明了边齐次与非边齐次k-边轨道地图存在有限指数m-边轨道极小覆盖（m≤k），并完成了n-叶覆盖的存在性与分类。

Vertex-transitive maps with Schläfli type {3, 7}

Among all equivelar vertex-transitive maps on a given closed surface S, the automorphism groups of maps with Schläfli types {3, 7} and {7, 3} allow the highest possible order. We describe a procedure to transform all such maps into 1- or 2-orbit maps, whose symmetry type has been previously studied. In so doing we provide a procedure to determine all vertex-transitive maps with Schläfli type {3, 7} which are neither regular or chiral. We determine all such maps on surfaces with Euler characteristic -1 \geq \c{hi} \geq -40.

Archimedean toroidal maps and their minimal almost regular covers

The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type $(4.8^2)$ is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equ…

环面地图与超地图的保向自同构群的度数

The degrees of the orientation-preserving automorphism groups of toroidal maps and hypermaps

本文研究高度对称的环面地图与超地图的保向自同构群的忠实传递置换表示，给出所有可能的度数列表，并利用GAP包corefreesub构造Schreier陪集图。

The Degrees Of Toroidal Regular Proper Hypermaps

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type $(3,3,3)$ that is a subgroup of index $2$ of the group of symmetries of a toroidal map of type $\{6,3\}$.

三次因子不变图的环商类型——交替情况

Cubic factor-invariant graphs of cycle quotient type -- the alternating case

本文研究了三次顶点传递图的交替类型，通过分类和确定2-因子的不变性，引入了一个新的无限图家族，扩展了广义佩特森图和蜂窝扭态图。