The obstructions for toroidal graphs with no K3,3’s (original) (raw)
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Forbidden minors and subdivisions for toroidal graphs with no K3,3's
Electronic Notes in Discrete Mathematics, 2005
Forbidden minors and subdivisions for toroidal graphs are numerous. In contrast, the toroidal graphs with no K 3,3 's have a nice explicit structure and short lists of obstructions. For these graphs, we provide the complete lists of four forbidden minors and eleven forbidden subdivisions.
Forbidden Minors for Projective Plane are Free-Toroidal or Non-Toroidal
Citeseer
Most of the definitions of the topological graph theory are the same as in [3]. The definitions concerning free minor closed classes are as in [5, 6, 7]. The definitions and denotations concerning forbidden subgraphs on projective plane are as in [3, 4]. Let us denote by APl,AP ,AT the classes of ...
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.
On the enumeration of a class of toroidal graphs
2018
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 36\{3^{6}\}36, 44\{4^{4}\}44, 63\{6^{3}\}63, 33,42\{3^{3}, 4^{2}\}33,42, 32,4,3,4\{3^{2}, 4, 3, 4\}32,4,3,4, 3,6,3,6\{3, 6, 3, 6\}3,6,3,6, 34,6\{3^{4}, 6\}34,6, 4,82\{4, 8^{2}\}4,82, 3,122\{3, 12^{2}\}3,122, 4,6,12\{4, 6, 12\}4,6,12, 3,4,6,4\{3, 4, 6, 4\}3,4,6,4. We know the classification of the maps of types 36\{3^{6}\}36, 44\{4^{4}\}44, 63\{6^{3}\}63 on the torus. In this article, we attempt to classify maps of types 33,42\{3^{3}, 4^{2}\}33,42, 32,4,3,4\{3^{2}, 4, 3, 4\}32,4,3,4, 3,6,3,6\{3, 6, 3, 6\}3,6,3,6, 34,6\{3^{4}, 6\}34,6, 4,82\{4, 8^{2}\}4,82, 3,122\{3, 12^{2}\}3,122, 4,6,12\{4, 6, 12\}4,6,12, 3,4,6,4\{3, 4, 6, 4\}3,4,6,4 on the torus.
Toroidal and Projective Cyclic Graphs
2017
All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown that the only finite groups with projective cyclic graphs are S 3 × Z 2 , D 14 , QD 16 and which all have toroidal cyclic graph too. Also, D 16 is characterized as the only finite group whose cyclic graph is toroidal but not projective
The closed 2-cell embeddings of 2-connected doubly toroidal graphs
Discrete Mathematics, 1995
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 closed 2-cell 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 doubly toroidal graph G has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover; i.e., G has a set of circuits containing every edge exactly twice.
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
Bull. ICA, 2021
Honeycomb toroidal graphs are trivalent Cayley graphs on generalized dihedral groups. We examine the two historical threads leading to these graphs, some of the properties that have been established, and some open problems.