Eulerian and even-face ribbon graph minors

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.     

On triconnected and cubic plane graphs on given point sets

Let S be a set of n4 points in general position in the plane, and let h<n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max{3n/2,n+h−1} edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n4 is even and hn/2+1. The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

Infinite paths in planar graphs IV,dividing cycles

Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 173–195, 2006     

Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple, reducible Venn diagram with five congruent ellipses. We show that this diagram is unique on the sphere and produces two different plane diagrams. This corrects some erroneous statements that started with John Venn more than a century ago in 1880 and have been repeated frequently by others ever since.     

Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

简单完整正则平面图

对简单完整正则平面图的特性和结构进行了分析和讨论 ,找出了简单完整正则平面图的可能的种类 .此外 ,对各种简单完整正则平面图的色数进行了求解 ,并用不同的方法给出了各个简单完整正则平面图的作色方案 .     

Entire choosability of near-outerplane graphs

It is proved that if G is a plane embedding of a K₄-minor-free graph with maximum degree Δ, then G is entirely 7-choosable if Δ≤4 and G is entirely (Δ+2)-choosable if Δ≥5; that is, if every vertex, edge and face of G is given a list of max{7,Δ+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K_2,3-minor-free graph or a -minor-free graph. As a special case this proves that the Entire Coluring Conjecture, that a plane graph is entirely (Δ+4)-colourable, holds if G is a plane embedding of a K₄-minor-free graph, a K_2,3-minor-free graph or a -minor-free graph.     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

On String Graph Limits and the Structure of a Typical String Graph

We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.     

1-平面图及其子类的染色

如果图G可以嵌入在平面上,使得每条边最多被交叉1次,则称其为1-可平面图,该平面嵌入称为1-平面图.由于1-平面图G中的交叉点是图G的某两条边交叉产生的,故图G中的每个交叉点c都可以与图G中的四个顶点(即产生c的两条交叉边所关联的四个顶点)所构成的点集建立对应关系,称这个对应关系为θ.对于1-平面图G中任何两个不同的交叉点c_1与c_2(如果存在的话),如果|θ(c_1)∩θ(c_2)|≤1,则称图G是NIC-平面图;如果|θ(c_1)∩θ(c_2)|=0,即θ(c_1)∩θ(c_2)=?,则称图G是IC-平面图.如果图G可以嵌入在平面上,使得其所有顶点都分布在图G的外部面上,并且每条边最多被交叉一次,则称图G为外1-可平面图.满足上述条件的外1-可平面图的平面嵌入称为外1-平面图.现主要介绍关于以上四类图在染色方面的结果.     

Products of unit distance graphs

Some graphs admit drawings in the Euclidean plane (k-space) in such a (natural) way, that edges are represented as line segments of unit length. We say that they have the unit distance property.The influence of graph operations on the unit distance property is discussed. It is proved that the Cartesian product preserves the unit distance property in the Euclidean plane, while graph union, join, tensor product, strong product, lexicographic product and corona do not. It is proved that the Cartesian product preserves the unit distance property also in higher dimensions.     

Proper 1-immersions of graphs triangulating the plane

In this paper we study what planar graphs are “rigid” enough such that they can not be drawn on the plane with few (1, 2, or 3) crossings per edge. A graph drawn on the plane is k$k$-immersed in the plane if each edge is crossed by at most k$k$ other edges. By a proper  k$k$-immersion of a graph we mean a k$k$-immersion of the graph in the plane such that there is at least one crossing point. We give a characterization (in terms of forbidden subgraphs) of 4-connected graphs which triangulate the plane and have a proper 1-immersion. We show that every planar graph has a proper 3-immersion.     

Facial unique-maximum colorings of plane graphs with restriction on big vertices

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova et al. (2018). We conclude the paper by proposing some problems.     

A sufficient condition on 3-colorable plane graphs without 5- and 6-circuits

In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- and 6-circuits and without triangles of distance less than 2, then G is 3-colorable.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

外平面图的弱边面染色

假设G是一个平面图.如果e1和e2是G中两条相邻边且在关联的面的边界上连续出现,那么称e1和e2面相邻.图G的一个弱边面κ-染色是指存在映射π:E∪F→{1,…,κ},使得任意两个相邻面、两条面相邻的边以及两个相关联的边和面都染不同的颜色.若图G有一个弱边面κ-染色,则称G是弱边面κ-可染的.平面图G的弱边面色数是指G是弱边面κ-可染的正整数κ的最小值,记为χef(G).2016年,Fabrici等人猜想:每个无环且无割边的连通平面图是弱边面5-可染的.本文证明了外平面图满足此猜想,即:外平面图是弱边面5-可染的.     

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.