Regular Embeddings Research Articles

Regular embeddings of complete bipartite graphs: classification and enumeration

The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where $n$ is a prime power, obtained in collaboration with Du, Kwak, Nedela and \v{S}koviera, together with results of It\^o, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups.

Classification of regular embeddings of n-dimensional cubes

An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Q n were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n=2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Q n for all n. In particular, we show that for all even n (=2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group S n such that σ fixes n, preserves the set of all pairs B i ={i,i+m} for 1≤i≤m, and induces the same permutation on this set as the permutation B i ↦ B f(i) for some additive bijection f:ℤ m →ℤ m . We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n.

A characterization of regular embeddings of [formula omitted]-dimensional cubes

One of the central problems in topological graph theory is the problem of the classification of graph embeddings into surfaces exhibiting a maximum number of symmetries. These embeddings are called regular. In particular, Du, Kwak and Nedela (2005) classified regular embeddings of n -dimensional cubes Q n for n odd. For even n Kwon has constructed a large family of regular embeddings with an exponential growth with respect to n . The classification was recently extended by J. Xu to numbers n = 2 m , where m is odd by showing that these embeddings coincide with the embeddings constructed by Kwon (2004) [21]. In the present paper we give a characterization of regular embeddings of Q n . We employ it to derive structural results on the automorphism groups of such embeddings as well as to construct a family of embeddings not covered by the Kwon embeddings.

Regular embeddings of the stationary tower and Woodin's maximality theorem

AbstractWe present Woodin's proof that if there exists a measurable Woodin cardinal δ then there is a forcing extension satisfying all sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in Vδ. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j: V → M with critical point such that M is countably closed in the forcing extension.

Regular embeddings of [formula omitted] where [formula omitted] is a power of 2. II: The non-metacyclic case

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 = X Y 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 .

Nonorientable regular embeddings of graphs of order [formula omitted

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex–edge–face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex–edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph G are, respectively, 2-cell embeddings of G as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order p q for two primes p and q ( p may be equal to q ) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for p ≠ q . In this paper, we shall classify the nonorientable regular embeddings of graphs of order p 2 where p is a prime so that a complete classification of regular embeddings of graphs of order p q for two primes p and q is obtained. All graphs in this paper are connected and simple.

Wilson's map operations on regular dessins and cyclotomic fields of definition

Dessins d'enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs.

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponentn– their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graphKn,n, wherenis an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphsKn,n[7] and generalises the analogous results for maps obtained in [9], partly using different methods.

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimension k. It is proved that any two regular (algebraic) embeddings X → ℝn are regularly equivalent, provided that n ⩾ 4k + 2.

Riemannian Manifold Learning

Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, i.e., how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low-dimensional space. Experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.

A classification of regular embeddings of hypercubes Q 2m with m odd

In this paper we give a classification of the orientable regular embeddings of Qn for n = 2m with m odd. Actually we prove that the known examples given by Kwon are the only regular maps of Qn when n = 2m (m odd).

Regular hamiltonian embeddings of [formula omitted] and regular triangular embeddings of [formula omitted

We give a group-theoretic proof of the following fact, proved initially by methods of topological design theory: Up to isomorphism, the number of regular hamiltonian embeddings of K n , n is 2 or 1 , depending on whether n is a multiple of 8 or not. We also show that for each n there is, up to isomorphism, a unique regular triangular embedding of K n , n , n .

An elementary {GIT} construction of the moduli space of stable maps

This paper provides an elementary construction of the moduli space of stable maps $\overline{M}_{0,0}(\mathbb{P}^r,d)$ as a sequence of weighted blow-ups along regular embeddings of a projective variety. This is a corollary to a more general GIT construction of $\overline{M}_{0,n}(\mathbb{P}^r,d)$ that places stable maps, the Fulton-MacPherson space $\mathbb{P}^1[n]$, and curves $\overline{M}_{0,n}$ into a single context.

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say n = 2 a p 1 a 1 p 2 a 2 ⋯ p k a k ( p 1 , p 2 , … , p k are distinct odd primes and a i > 0 for each i ⩾ 1 ) , there are t distinct reflexible regular embeddings of the complete bipartite graph K n , n up to isomorphism, where t = 1 if a = 0 , t = 2 k if a = 1 , t = 2 k + 1 if a = 2 , and t = 3 · 2 k + 1 if a ⩾ 3 . And, there are s distinct self-Petrie dual regular embeddings of K n , n up to isomorphism, where s = 1 if a = 0 , s = 2 k if a = 1 , s = 2 k + 1 if a = 2 , and s = 2 k + 2 if a ⩾ 3 .

Galois action on families of generalised Fermat curves

It is well known that the complete bipartite graphs K n , n occur as dessins d'enfants on the Fermat curves of exponent n. However, there are many more curves having K n , n as the underlying graph of their dessins, even if we require the strongest regularity condition that the graphs define regular maps on the underlying Riemann surfaces. For odd prime powers n these maps have recently been classified [G.A. Jones, R. Nedela, M. Škoviera, Regular embeddings of K n , n where n is an odd prime power, European J. Combin., in press]; they fall into certain families characterised by their automorphism groups. In the present paper we show that these families form Galois orbits. We determine the minimal field of definition of the corresponding curves, and in easier cases also their defining equations.

Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

For G any finite group, the left and right regular representations λ, respectively ρ of G into Perm ( G ) map G into InHol ( G ) = ρ ( G ) ⋅ Inn ( G ) . We determine regular embeddings of G into InHol ( G ) modulo equivalence by conjugation in Hol ( G ) by automorphisms of G, for groups G that are semidirect products G = Z h ⋊ Z k of cyclic groups and have trivial centers. If h is the power of an odd prime p, then the number of equivalence classes of regular embeddings of G into InHol ( G ) is equal to twice the number of fixed-point free endomorphisms of G, and we determine that number. Each equivalence class of regular embeddings determines a Hopf Galois structure on a Galois extension of fields L / K with Galois group G. We show that if H 1 is the Hopf algebra that gives the standard non-classical Hopf Galois structure on L / K , then H 1 gives a different Hopf Galois structure on L / K for each fixed-point free endomorphism of G.

Regular embeddings ofKn,nwherenis a power of 2. I: Metacyclic case

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 Kn,n, where n=2e. The method involves groups G which factorize 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 Kn,n based on that for G. We prove that given n=2e (for e≥3), there are, up to map isomorphism, exactly 2e−2+4 regular embeddings of Kn,n. Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.

Regular embeddings ofKn,nwherenis an odd prime power

We show that if n=pe where p is an odd prime and e≥1, then the complete bipartite graph Kn,n has pe−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 Kn,n has at least n/∏p|np orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p≡1mod(q).

Non-existence of nonorientable regular embeddings of n-dimensional cubes

By a regular embedding of a graph K into a surface we mean a two-cell embedding of K into a compact connected surface with the automorphism group acting regularly on flags. Regular embeddings of the n-dimensional cubes Q n into orientable surfaces exist for any positive integer n. In contrast to this, we prove the nonexistence of nonorientable regular embeddings of Q n for n > 2 .

Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n -dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q n into nonorientable surfaces exist for any positive integer n > 2 . In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807–823] presented a construction giving for each solution of the congruence e 2 ≡ 1 ( mod n ) a regular embedding M e of the hypercube Q n into an orientable surface. It was conjectured that all regular embeddings of Q n into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q n into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n.