Regular orientable imbeddings of complete graphs

European Journal of Combinatorics, 2007

We show that if n = p e where p is an odd prime and e ≥ 1, then the complete bipartite graph K n,n has p e−1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n, n} and genus (n − 1)(n − 2)/2; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n. We deduce that if n is odd then K n,n has at least n/ p|n p orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p ≡ 1 mod (q).

European Journal of Combinatorics, 2005

The merged Johnson graph J (n, m) I is the union of the distance i graphs J (n, m) i of the Johnson graph J (n, m) for i ∈ I , where ∅ = I ⊆ {1,. .. , m} and 2 ≤ m ≤ n/2. We find the automorphism groups of these graphs, and deduce that their only regular embedding in an orientable surface is the octahedral map on the sphere for J (4, 2) 1 , and that they have just six non-orientable regular embeddings. This yields classifications of the regular embeddings of the line graphs L(K n) = J (n, 2) 1 of complete graphs, their complements L(K n) = J (n, 2) 2 , and the odd graphs O m+1 = J (2m + 1, m) m .

Graphs and Combinatorics, 2006

A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B n , a closed-end ladder L n or a Ringel ladder R n . A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an underlying bouquet B n in this paper.

Journal of the London Mathematical Society, 1999

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H v is the cyclic subgroup of index 2 in G v. An analogous result is proved for orientably-regular embeddings.

Ars Mathematica Contemporanea

Regular embeddings of cycles with multiple edges have been reappearing in the literature for quite some time, both in and outside topological graph theory. The present paper aims to draw a complete picture of these maps by providing a detailed description, classification, and enumeration of regular embeddings of cycles with multiple edges on both orientable and non-orientable surfaces. Most of the results have been known in one form or another, but here they are presented from a unique viewpoint based on finite group theory. Our approach brings additional information about both the maps and their automorphism groups, and also gives extra insight into their relationships.

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.

European Journal of Combinatorics, 2018

A map is said to be even-closed if all of its automorphisms act like even permutations on the vertex set. In this paper the study of even-closed regular maps is approached by analysing two distinguished families. The first family consists of embeddings of a wellknown family of graphs on distinct orientable surfaces, whereas in the second family we consider all graphs having orientable-regular embeddings on a particular surface. In particular, the classification of even-closed orientable-regular embeddings of the complete bipartite graphs K n,n and classification of even-closed orientableregular maps on the torus are given.

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

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.

2009

In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2-edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.