The obstructions for toroidal graphs with no K3,3’s (original) (raw)
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.
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.
On H-antimagic decomposition of toroidal grids and triangulations
AKCE International Journal of Graphs and Combinatorics
Let G ¼ ðV, EÞ be a finite simple graph with p vertices and q edges. A decomposition of a graph G into isomorphic copies of a graph H is called (a, d)-H-antimagic if there is a bijection f : V [ E ! f1, 2, :::, p þ qg such that for all subgraphs H 0 isomorphic to H in the decomposition of G, the sum of the labels of all the edges and vertices belonging to H 0 constitutes an arithmetic progression with the initial term a and the common difference d. When f ðVÞ ¼ f1, 2, :::, pg, then G is said to be super (a, d)-H-antimagic and if d ¼ 0 then G is called H-supermagic. In the paper we examine the existence of such labelings for toroidal grids and toroidal triangulations.
Labeled K2, t Minors in Plane Graphs
Journal of Combinatorial Theory, Series B, 2002
Let G be a 3-connected planar graph and let U ⊆ V (G). It is shown that G contains a K 2,t minor such that t is large and each vertex of degree 2 in K 2,t corresponds to some vertex of U if and only if there is no small face cover of U. This result cannot be extended to 2-connected planar graphs.
Embedding Graphs Containing K5-Subdivisions
Ars Combinatoria, 2002
Given a non-planar graph G with a subdivision of K5 as a subgraph, we can either transform the K5-subdivision into a K3,3-subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equiva- lence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple
Nonplanarity of unit graphs and classification of the toroidal ones
Pacific Journal of Mathematics, 2014
The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal.
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:
The 3-choosability of plane graphs of girth 4
Discrete Mathematics, 2005
A set S of vertices of the graph G is called k-reducible if the following is true: G is k-choosable if and only if G − S is k-choosable. A k-reduced subgraph H of G is a subgraph of G such that H contains no k-reducible set of some specific forms. In this paper, we show that a 3-reduced subgraph of a non-3-choosable plane graph G contains either adjacent 5-faces, or an adjacent 4-face and kface, where k 6. Using this result, we obtain some sufficient conditions for a plane graph to be 3-choosable. In particular, if G is of girth 4 and contains no 5-and 6-cycles, then G is 3-choosable.
Honeycomb toroidal graphs are Cayley graphs
Information Processing Letters, 2009
There is a particular family of trivalent vertex-transitive graphs that have been called generalized honeycomb tori by some and brick products by others. They have been studied as hexagonal embeddings on the torus as well. We show that all these graphs are Cayley graphs on generalized dihedral groups.
Some structural properties of planar graphs and their applications to 3-choosability
Discrete Mathematics, 2012
In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph G is 3-choosable if it is satisfied one of the following conditions: (1) each vertex x is neither incident to cycles of lengths 4, 9, ix with ix ∈ {5, 7, 8}, nor incident to 6-cycles adjacent to a 3-cycle. (2) each vertex x is not incident to cycles of lengths 4, 7, 9, ix with ix ∈ {5, 6, 8}. This work implies five results already published [13, 3, 7, 12, 4].