Regular embeddings of cycles with multiple edges revisited

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.

Journal of Combinatorial Theory, Series B, 1985

This paper classifies the regular imbeddings of the complete graphs K,, in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p', his examples being Cayley maps based on the finite field F= GF(n). We show that these are the only examples, and that there are q5(n-1)/e isomorphism classes of such maps (where 4 is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Qn-r(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations Hi, and we determined for which n they are reflexible or self-dual.

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.

Journal of Combinatorial Theory, Series B, 1996

European Journal of Combinatorics, 2007

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs K n,n , where n = 2 e . The method involves groups G which factorise as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of K n,n based on that for G. We prove that given n = 2 e , e ≥ 3 there are, up to map isomorphism, exactly 2 e−2 +4 regular embeddings of K n,n . Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.

European Journal of Combinatorics, 2011

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.

Graphs and Combinatorics, 1999

If G and H are vertex-transitive graphs, then the framing number f r(G, H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates f r(C m , C n ) for m < n. We show first that f r(C m , C n ) ≥ n + 2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that f r(C m , C n ) is approximately the minimum of n − m + 2 √ n and n + n/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.

European Journal of Combinatorics, 2010

The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K n,n , where n = 2 e. The method involves groups G which factorise as a product G = XY of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n = 2 e , e ≥ 3, there are up to map isomorphism exactly four regular embeddings of K n,n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n = 4.

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.