Cayley maps

Discrete Mathematics, 1994

A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at every vertex are identical. In this series we consider two types of such maps: the balanced and antibalanced

University of Paris, 1969

Discrete Mathematics, 2007

The class of t-balanced Cayley maps [J. Martino, M. Schultz, Symmetrical Cayley maps with solvable automorphism groups, abstract in SIGMAC '98, Flagstaff, AR, 1998] is a natural generalisation of balanced and antibalanced Cayley maps introduced and studied by Širáň and Škoviera [Regular maps from Cayley graphs II: antibalanced Cayley maps, Discrete Math. 124 (1994) 179-191; Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202]. The present paper continues this study by investigating the distribution of inverses, automorphism groups, and exponents of t-balanced Cayley maps. The methods are based on the use of t-automorphisms of groups with sign structure which extend the notion of an antiautomorphism crucial for antibalanced Cayley maps. As an application, a new series of nonstandard regular embeddings of complete bipartite graphs K n,n is constructed for each n divisible by 8.

Ars Mathematica Contemporanea

This paper explains some facts probably known to experts and implicitely contained in the literature about dessins d'enfants but which seem to be nowhere explicitely stated. The 1skeleton of every regular Cori hypermap is the Cayley graph of its automorphism group, embedded in the underlying orientable surface. Conversely, every Cayley graph of a finite two-generator group has an embedding as the 1-skeleton of a regular hypermap in the Cori representation. For non-regular hypermaps there is an analogous correspondence with Schreier coset diagrams.

2009

keywords: Cayley graphs, groups, visualizations, proofs.

2006

A map is a connected topological graph Γ 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 Γ = Cay(G : S) with AutΓ ∼ = G × H on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of Γ 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.

Filomat

In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian 2-group, or its Sylow 2-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.

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: General Topology, 2019

In this article we discuss a connection between two famous constructions in mathematics: a Cayley graph of a group and a (rational) billiard surface. For each rational billiard surface, there is a natural way to draw a Cayley graph of a dihedral group on that surface. Both of these objects have the concept of "genus" attached to them. For the Cayley graph, the genus is defined to be the lowest genus amongst all surfaces that the graph can be drawn on without edge crossings. We prove that the genus of the Cayley graph associated to a billiard surface arising from a triangular billiard table is always zero or one. One reason this is interesting is that there exist triangular billiard surfaces of arbitrarily high genus , so the genus of the associated graph is usually much lower than the genus of the billiard surface.