The closed 2-cell embeddings of 2-connected doubly toroidal graphs (original) (raw)

Separating Cycles in Doubly Toroidal Embeddings

2003

We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle that separates the surface into two pieces. As a special case, every triangulation of the double torus in which every noncontractible cycle has length at least 4 has a noncontractible cycle that separates the surface into two pieces.

On the flexibility of toroidal embeddings

Journal of Combinatorial Theory, Series B, 2008

Two embeddings Ψ 1 and Ψ 2 of a graph G in a surface Σ are equivalent if there is a homeomorphism of Σ to itself carrying Ψ 1 to Ψ 2 . In this paper, we classify the flexibility of embeddings in the torus with representativity at least 4. We show that if a 3-connected graph G has an embedding Ψ in the torus with representativity at least 4, then one of the following holds:

ON CHIRALITY OF TOROIDAL EMBEDDINGS OF POLYHEDRAL GRAPHS

On chirality of toroidal embeddings of polyhedral graphs, 2017

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalising a theorem of Simon [6]. topological graphs; knots and links; chirality; topology and chemistry; templating on a toroidal substrate

Closed 2-cell embeddings of graphs with no -minors

Discrete Mathematics, 2001

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.

Embeddings of Small Graphs on the Torus

2003

Embeddings of graphs on the torus are studied. All 2-cell embeddings of the vertex-transitive graphs on 12 vertices or less are constructed. Their automorphism groups and dual maps are also constructed. A table of em- beddings is presented. 1. Toroidal Graphs

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.

Closed 2-cell embeddings of 4 cross-cap embeddable graphs

Discrete Mathematics, 1996

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a circuit in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. A graph is called k cross-cap embeddable if it can be embedded in the non-orientable surface of k cross-caps. In this paper, we prove that every 2-connected 4 cross-cap embeddable graph G has a closed 2-cell embedding in some surface. As a corollary, G has a cycle double cover, i.e., G has a set of circuits containing every edge exactly twice.

Closed 2-cell embeddings of graphs with no V8-minors

Discrete Mathematics, 2001

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.

On enumeration of a class of toroidal graphs

2013

We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types {3^6}, {4^4}, {6^3}, {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4}. We know the classification of the maps of types {3^6}, {4^4}, {6^3} on the torus. In this article, we attempt to classify maps of types {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4} on the torus.

The obstructions for toroidal graphs with no K3,3’s

2005

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3-subdivisions that coincide with the toroidal graphs with no K3,3-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no K3,3’s and prove that the lists are sufficient.