Enumerating 222-cell imbeddings of connected graphs (original) (raw)

A.T.White, Enumeration 2-cell imbeddings of connected graphs,Proc.Amer.Math.Soc

2013

ABSTRACT. A systematic approach is developed for enumerating congruence classes of 2-cell imbeddings of connected graphs on closed orientable 2-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration. 1. Introduction. An

Regular orientable imbeddings of complete graphs

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.

The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration

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.

Genus embeddings for some complete tripartite graphs

Discrete Mathematics, 1976

The vu&age graph construction of Gross is extenSccf to the case where the baw graph is non-orkntably embedded. An easily applied criterion is established for determining the orientability character of the derived embedding. These methods are then applied to derive both orientable ijnd non-orientabte genus embeddings for some families of complete tripartite graphs.

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

Discrete Mathematics, 2012

A genus-g map 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.

Obstructions for two-vertex alternating embeddings of graphs in surfaces

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

The imbedding index of a graph

Journal of Combinatorial Theory, Series B, 1979

The natural extension of MacLane's combinatorial approach to planar imbeddings is seen to yield a combinatorial formulation of imbedding of a graph in a pseudosurface. This leads to a combinatorially defined parameter for all graphs, called the imbedding index. A generalization of the Heaword inequahty is then proved for this parameter.

The enumeration of planar graphs via Wick's theorem

Advances in Mathematics, 2009

A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type partition function. Some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover. We express the number of the graphs with a fixed directed cycle double cover as the Gaussian matrix integral of an Ihara-Selberg-type function.

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

Acta Mathematica Sinica-english Series, 2005

A graph is called a semi–regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi–regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation–preserving automorphisms of maps on surfaces (orientable and non–orientable) or just orientable surfaces with a given underlying semi–regular graph Γ are determined. Formulas for the numbers of non–equivalent embeddings of this kind of graphs on surfaces (orientable, non–orientable or both) are established, and especially, the non–equivalent embeddings of circulant graphs of a prime order on orientable, non–orientable and general surfaces are enumerated.

A common cover of graphs and 2-cell embeddings

Journal of Combinatorial Theory, Series B, 1986

Let G and H be finite graphs with equal uniform degree refinements. Their finite common covering graph G 0 H is constructed. It is shown that G, H, and G 0 H can be 2-cell embedded in orientable surfaces M, N and S", respectively, in such a way that the graph covering projections G 0 H + G and G 0 H + H extend to branched coverings M + S-+ N of the surfaces. Additional properties of G 0 H are used to obtain some nontrivial consequences about coverings of some planar graphs.