Enumeration of 3-regular one-face maps on orientable or non-orientable surface up to all symmetries

We obtain explicit formulas for enumerating $3$-regular one-face maps on orientable and non-orientable surfaces of a given genus $g$ up to all symmetries. We use recent analytical results obtained by Bernardi and Chapuy for counting rooted precubic maps on non-orientable surfaces together with more widely known formulas for counting precubic maps on orientable surfaces. To take into account all symmetries we use a result of Krasko and Omelchenko that allows to reduce this problem to the problem of counting rooted quotient maps on orbifolds.Comment: arXiv admin note: substantial text overlap with arXiv:1712.1013

Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus

In this work for the first time we enumerate unlabelled maps on orientable genus $g$ surfaces with respect to all homeomorphisms, including both orientation-preserving and orientation-reversing. We show that in the latter case as an intermediate step one has to enumerate rooted maps of a special kind (quotient maps) on orientable and non-orientable surfaces possibly having a boundary and a certain number of branch points. In this work we develop a special technique for enumerating such maps

Automorphisms and Enumeration of Maps of Cayley Graph of a Finite Group

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph $\Gamma={\rm Cay}(G:S)$ with ${\rm Aut} \Gamma\cong G\times H$ on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of $\Gamma$ on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric groups, groups generated by 3 involutions and abelian groups on orientable or non-orientable surfaces.Comment: 17 pages with 1 figur

The Number of Complete Maps on Surfaces

A map is a connected topological graph cellularly embedded in a surface and a complete map is a cellularly embedded complete graph in a surface. In this paper, all automorphisms of complete maps of order n are determined by permutations on its vertices. Applying a scheme for enumerating maps on surfaces with a given underlying graph, the numbers of unrooted complete maps on orientable or non-orientable surfaces are obtained.Comment: 21 pages with 2 figure

Enumeration of rr-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps

The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, projective plane and the Klein bottle. We also present the results of enumerating some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root semi-edges. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then using these results we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation --- so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms --- so-called unsensed maps

Enumeration of rr-regular Maps on the Torus. Part II: Enumeration of Unsensed Maps

The second part of the paper is devoted to enumeration of $r$-regular toroidal maps up to all homeomorphisms of the torus (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, annulus and the M\"obius band. Using $3$- and $4$-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed $r$-regular maps on the torus for various $r$

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group in the group of outer automorphisms of the symmetry group of a tiling. We explicitly give descriptions of certain subgroups of mapping class groups and of tilings as decorations on orbifolds, namely those that are commensurate with the Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this explicit description to give an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations, outlining how the approach yields an unambiguous enumeration

A new view of combinatorial maps by Smarandache's notion

On a geometrical view, the conception of map geometries are introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surface. Results convinced one that map geometries are Smarandache geometries and their enumertion are obtained. Open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are also presented in this paper.Comment: 19page

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

Asymptotic enumeration of labelled graphs with a given genus

We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled graphs of genus g (1-connected, 2-connected, and 3-connected graphs.)Comment: 28 pages; corrected, clarified & simplified presentatio