2-cell Embedding Research Articles

The maximum genus of the generalized Petersen Graph, GP (<em>n, k</em>) for the cases <em>k</em> = 1, 2

In Topological graph theory, the maximum genus of graphs has been a fascinating subject. For a simple connected graph G, the maximum genus γM(G) is the largest genus of an orientable surface on which G has a 2-cell embedding. γM(G) has the upper bound, γM(G)≤[β/2], where β(G) denotes the Betti number and G is said to be upper embeddable if the equality holds. In this study, the maximum genus of GP(n, k) is established as γM(GP(n,k))=[(n+1)/2] for k = 1 and k = 2 by proving the upper embeddability of generalized Petersen graph, GP(n, k) for the cases k = 1 and k = 2. The proof is done by obtaining spanning trees T and examining the components in the edge complements GP(n, k)\T for the cases k = 1 and k = 2 of GP(n, k).

Regular dessins with underlying graphs [formula omitted]: The metacyclic case

A dessin is a 2-cell embedding of a connected 2-colored bipartite graph into an orientable closed surface. A dessin is regular if its group of automorphisms, preserving both color and orientation, acts transitively on its edge set. Leveraging the theory of metacyclic 2-groups, we undertake the classification and enumeration of regular dessins with complete bipartite underlying graphs of the form K2e,2f, while imposing the additional criterion of metacyclic automorphism groups. Our findings extend and generalize the earlier work of Du et al. (2007) [5].

Embedding [formula omitted] and [formula omitted] on the double torus

The Kuratowski graphs K3,3 and K5 characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by Mull (1999) and Mull et al. (2008), using Burnside’s Lemma and automorphism groups of K3,3 and K5, without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of K3,3, and 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one-face embeddings of K5 on the triple torus are also found, using exhaustive search.

Random 2-cell embeddings of multistars

Random 2-cell embeddings of a given graphGGare obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces,E[FG]\mathbb {E}[F_G], of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar withnnnonleaf edges lies in an interval of length2/(n+1)2/(n+1)centered at the expected number of faces of annn-edge dipole. This allows us to derive bounds onE[FG]\mathbb {E}[F_G]for any given graphGGin terms of vertex degrees. We conjecture thatE[FG]≤O(n)\mathbb {E}[F_G]\le O(n)for any simplenn-vertex graphGG.

Regular Cayley maps for dihedral groups

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut∘M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut∘M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M(n,r), one for odd n and one for even n, where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families Mi(n,r), 1≤i≤5, are derived, where 2n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map Mi(n,r), we determine its valence and covalence, and also describe the structure of the group Aut∘Mi(n,r). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms.

Z-Knotted Triangulations of Surfaces

A zigzag in a map (a 2-cell embedding of a connected graph in a connected closed 2-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: (1) any two consecutive edges lie on the same face and have a common vertex, (2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third one. A map is z-knotted if it contains a single zigzag. Such maps are closely related to the Gauss code problem and have nice homological properties. We show that every triangulation of a connected closed 2-dimensional surface admits a z-knotted shredding.

Complete regular dessins of odd prime power order

A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine and enumerate the isomorphism classes of regular dessins with complete bipartite underlying graphs of odd prime power order.

Enumerating Unlabelled Embeddings of Digraphs

AbstractA 2-cell embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles Bn, directed dipoles OD2n and for a class of regular tournaments T2n+1.

Classification of reflexible Cayley maps for dihedral groups

A regular map M is a 2-cell embedding of a connected graph into an orientable surface such that the group of all orientation-preserving automorphisms of the embedding acts transitively on the set of all incident vertex-edge pairs called arcs. Such a map M is called a regular Cayley map for the finite group G if M is the embedding of a Cayley graph C(G,S) such that G induces a vertex-transitive group of map automorphisms preserving orientation. In addition, if there is an orientation-reversing automorphism, the map is called reflexible. In this paper, we classify all reflexible Cayley maps for dihedral groups.

Generating Maps on Surfaces

We describe procedures for generating all 2-cell embedded simple graphs with up to a fixed number of vertices on a given surface. We also modify these procedures to generate closed 2-cell embeddings and polyhedral embeddings. We give results of computer implementations of these procedures for seven surfaces: the sphere, the torus, the double torus, the projective plane, the Klein bottle, the triple cross surface, and the quadruple cross surface.

Partial duality and closed 2-cell embeddings

In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell.

Non-abelian almost totally branched coverings over the platonic maps

A map is a 2-cell embedding of a connected graph into a closed surface. A map is orientable if the supporting surface is orientable. An orientable map is regular if its group of orientation-preserving automorphisms acts transitively on the darts. Using an equivalent algebraic description of regular maps and their coverings, we employ the theory of group extensions to classify the almost totally branched coverings of the platonic maps with non-abelian covering transformation groups, generalising the results of Hu, Nedela and Wang.

On Embeddings of Circulant Graphs

A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).

Chamfering operation on k-orbit maps

A map, as a 2-cell embedding of a graph on a closed surface, is called a k -orbit map if the group of automorphisms (or symmetries) of the map partitions its set of flags into k orbits. Orbanić, Pellicer and Weiss studied the effects of operations as medial and truncation on k -orbit maps. In this paper we study the possible symmetry types of maps that result from other maps after applying the chamfering operation and we give the number of possible flag-orbits that has the chamfering map of a k -orbit map, even if we repeat this operation t times.

Doubly Even Orientable Closed 2-Cell Embeddings of the Complete Graph

For all $m\geq 1$ and $k\geq 2$, we construct closed 2-cell embeddings of the complete graph $K_{8km+4k+1}$ with faces of size $4k$ in orientable surfaces. Moreover, we show that when $k\geq 3$ there are at least $(2m-1)!/2(2m+1)=2^{2m\text{log}_2m-\mathrm{O}(m)}$ nonisomorphic embeddings of this type. We also show that when $k=2$ there are at least $\frac14 \pi^{\frac12}m^{-\frac{5}{4}}\left(\frac{4m}{e^2}\right)^{\sqrt{m}}(1-\mathrm{o}(1))$ nonisomorphic embeddings of this type.

Nowhere-Zero 3-Flows in Signed Graphs

Tutte observed that every nowhere-zero $k$-flow on a plane graph gives rise to a $k$-vertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph $G$ has a face-$k$-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero $k$-flow. However, if the surface is nonorientable, then a face-$k$-coloring corresponds to a nowhere-zero $k$-flow in a signed graph arising from $G$. Graphs embedded in orientable surfaces are therefore a special case that the corresponding signs are all positive. In this paper, we prove that if an 8-edge-connected signed graph admits a nowhere-zero integer flow, then it has a nowhere-zero 3-flow. Our result extends Thomassen's 3-flow theorem on 8-edge-connected graphs to the family of all 8-edge-connected signed graphs. And it also improves Zhu's 3-flow theorem on 11-edge-connected signed graphs.

Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings i: X → S and j: X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that ih = γj. A 2-cell embedding i : X → S of a graph X into a closed orientable surface S is described combinatorially by a pair (X; ρ) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S. The mirror image of a map (X; ρ) is themap (X; ρ −1), and one of the corresponding embeddings is called the mirror image of the other. A 2-cell embedding of X is reflexible if it is congruent to its mirror image. Mull et al. [Proc Amer Math Soc, 1988, 103: 321–330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces. In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces, and apply it to the complete graphs, the bouquets of circles, the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible (called chiral) embeddings.

Character-theoretic techniques for near-central enumerative problems

The centre of the symmetric group algebra C[Sn] has been used successfully for studying important problems in enumerative combinatorics. These include maps in orientable surfaces and ramified covers of the sphere by curves of genus g, for example. However, the combinatorics of some equally important Sn-factorization problems forces k elements in {1,…,n} to be distinguished. Examples of such problems include the star factorization problem, for which k=1, and the enumeration of 2-cell embeddings of dipoles with two distinguished edges associated with Berenstein–Maldacena–Nastase operators in Yang–Mills theory, for which k=2. Although distinguishing these elements obstructs the use of central methods, these problems may be encoded algebraically in the centralizer of C[Sn] with respect to the subgroup Sn−k. We develop methods for studying these problems for k=1, and demonstrate their efficacy on the star factorization problem. In a subsequent paper, we consider a special case of the above dipole problem by means of these techniques.

Classification of nonorientable regular embeddings of Hamming graphs

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the Hamming graph H(d,n). We show that there exists such an embedding if and only if n=2 and d=2, or n=3 or 4 and d≥1, or n=6 and d=1 or 2. We also give constructions and descriptions of these embeddings.

Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices

A genus-gmap is a 2-cell embedding of a connected graph on a closed, orientable surface of genus g without boundary, that is, a sphere with g handles. Two maps are equivalent if they are related by a homeomorphism between their embedding surfaces that takes the vertices, edges and faces of one map into the vertices, edges and faces, respectively, of the other map, and preserves the orientation of the surfaces. A map is rooted if a dart of the map–half an edge–is distinguished as its root. Two rooted maps are equivalent if they are related by a homeomorphism that has the above properties and that also takes the root of one map into the root of the other. By counting maps, rooted or unrooted, we mean counting equivalence classes of those maps.To count unrooted genus-g maps, we first needed to count rooted maps of every genus up to g. A recursively-defined generating function for counting rooted maps of arbitrary genus by number of faces and vertices was found by Didier Arquès and the second author, and numerical values were obtained for g≤3. The first two authors jointly extended the solution of this recursion up to g=5. Valery Liskovets used quotient maps and Tutte’s enumeration formula for rooted genus-0 maps to count unrooted genus-0 maps by number of edges. The third author and Roman Nedela generalized Liskovets’ method to count unrooted maps of genus 1, 2 and 3 by number of edges.In this paper, we describe the above-mentioned previously-obtained results and present the following new ones. The first author wrote an optimized program in C to extend the solution of the Arquès–Giorgetti recurrence, and thus the enumeration of rooted maps by number of edges and vertices, up to g=10. The second and third authors extended the enumeration of rooted and unrooted maps by number of edges up to genus 11 using tables computed and published on a web site by Ján Karabás, a student of Nedela. Using Liskovets’ refinement of the Mednykh–Nedela method and the tables of numbers of rooted maps, the first author counted unrooted maps of genus up to 10 by number of edges and vertices.