Some Semi-equivelar Maps of Euler Characteristics-2

2020

If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types then the map is said to be a Semi-equivelar map. In this article we classify all semi-equivelar and vertex transitive maps on the surface of Euler genus 3, i.e., on the surface of Euler characteristic -1.

Mathematica Slovaca

Semi-equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.

2021

If the face-cycles at all the vertices in a map are of the same type, then the map is said to be a semi-equivelar map. Automorphism (symmetry) of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. The set of all symmetries forms the symmetry group. In this article, we discuss the maps’ symmetric groups on higher genus surfaces. In particular, we show that there are at least 39 types of the semi-equivelar maps on the surface with Euler char. −2m,m ≥ 2 and the symmetry groups of the maps are isomorphic to the dihedral group or cyclic group. Further, we prove that these 39 types of semi-equivelar maps are the only types on the surface with Euler char. −2. Moreover, we know the complete list of semi-equivelar maps (up to isomorphism) for a few types. We extend this list to one more type and can classify others similarly. We skip this part in this article. MSC 2010 : 52C20, 52B70, 51M20, 57M60.

arXiv (Cornell University), 2020

A vertex v in a map M has the face-sequence (p n1 1 . . . . .p n k k ), if there are consecutive n i numbers of p i -gons incident at v in the given cyclic order, for 1 ≤ i ≤ k. A map M is called a semi-equivelar map if each of its vertex has same face-sequence. Doubly semi-equivelar maps are a generalization of semi-equivelar maps which have precisely 2 distinct face-sequences. In this article, we enumerate the types of doubly semi-equivelar maps on the plane and torus which have combinatorial curvature 0. Further, we present classification of doubly semi-equivelar maps on the torus and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs {(3 6 ), (3 3 .4 2 )} and {(3 3 .4 2 ), (4 4 )}.

Electronic Notes in Discrete Mathematics, 2007

The problem of classifying orientable vertex-transitive maps on a surface with genus two is considered. We construct and classify all simple orientable vertex-transitive maps, with face width at least 3 which can be viewed as generalisations of classical Archimedean solids. The proof is computer-aided. The developed method applies to higher genera as well.

Journal of Mathematics

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

Journal of Combinatorial Theory, Series B, 2012

In an earlier paper by A. Breda, R. Nedela and J. Širáň, a classification was given of all regular maps on surfaces of negative prime Euler characteristic. In this article we extend the classification to surfaces with Euler characteristic −3p (equivalently, to non-orientable surfaces of genus 3p + 2) for all odd primes p.

2011

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automor-phisms imply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications. The first edition of this book is in fact consisting of my post-doctoral reports in Chi-nese Academy of Sciences in 2005, not self-contained and not suitable as a textbook for graduate students. Many friends of mine suggested me to extend it to a textbook for graduate students in past years. That is the initial motivation of this editi...

Journal for Geometry and Graphics

The author defines what a uniform (or Platonic) map of type {p,q} on a surface is and enumerates all such maps on the Klein bottle. Uniform maps on the sphere are five in number and they correspond to the five Platonic solids. The parameters p and q in each of these maps stand, respectively, for the number of edges in each face and the number of edges emanating from each vertex of the corresponding Platonic solid. Uniform maps on the torus were determined by W. Kurth in [Discrete Math. 61, 71–83 (1986; Zbl 0594.05037)]. Thus by determining all the uniform maps on the Klein bottle (which the author shows to be necessarily of the types {4,4}, {3,6}, and {6,3}), the paper under review completes this kind of enumeration for all surfaces of non-negative Euler characteristic.

We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus p + 2 where p is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus p + 2 where p is a prime such that p ≡ 1 (mod 12) and p = 13.