Minimum genus embeddings of the complete graph

In this paper,the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied.There are three approaches to solve this problem.The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths;the second approach is to find a current assignment of the current graph by the theory of current graph;the third approach is to find exponentially many embedding(or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph.According to this three approaches,we can construct exponentially many minimum genus embeddings of complete graph K_(12s+8) in orientable surfaces,which show that there are at least 10/3×(200/9)~s distinct minimum genus embeddings for K_(12s+8) in orientable surfaces.We have also proved that K_(12s+8) has at least 10/3×(200/9)~s distinct minimum genus embeddings in non-orientable surfaces.     

不可定向曲面上的最大亏格嵌入和最小亏格嵌入

研究了不可定向曲面上最大亏格嵌入的估计数,得到了几类图的指数级不可定向最大亏格嵌入的估计数的下界.利用电流图理论,证明了完全图K_(12s)在不可定向曲面上至少有2~(3s-1)个最小亏格嵌入;完全图K_(12s+3)在不可定向曲面上至少有2~(2s)个最小亏格嵌入;完全图K_(12s+7)在不可定向曲面上至少有2~(2s+1)个最小亏格嵌入.     

Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover.Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.     

多重圈梯图在射影平面上的嵌入个数

图在不同亏格曲面上的嵌入个数常常有相关关系,因此,分析一些图类在小亏格曲面上的嵌入个数对最终确定图的亏格分布和完全亏格分布有着重要意义,本文利用嵌入的联树模型得出了多重圈梯图在射影平面上的嵌入个数.     

Développements récents de la théorie des graphies

Stimulated by its numerous applications, the theory of graphs continuously progresses in various domains. The concept of graph in itself varies in accordance with authors and problems considered: it belongs to the logic of sets as well as to combinatorial topology. Questions may be grouped into three main classes: the study of characteristic properties (invariants such as connectivity, existence of Hamiltonian circuits, Ramsey numbers); problems of embedding on surfaces (planarity, genus, thickness) and of graph-coloring; and enumeration of given classes of graphs, a peculiarly difficult field of investigation. Hypergraphs present a generalization of graphs from a set theoretic point of view. In spite of its rapid growth, graph theory has not yet reached its deep unity.     

Strong embeddings of minimum genus

A “folklore conjecture, probably due to Tutte” (as described in [P.D. Seymour, Sums of circuits, in: Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, 1977), Academic Press, 1979, pp. 341-355]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. Sporadic counterexamples to this conjecture have been known since the late 1970s. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing a plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.     

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

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 Möbius 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 classification of internally-4-connected graphs with no V₈-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.     

两类四正则图的完全亏格分布

一个图G的完全亏格多项式表征了图G的亏格(可定向,不可定向)分布情况.本文利用刘彦佩提出的嵌入的联树模型,得出了两类新的四正则图的完全亏格多项式,并推导出已有结果的两类图的完全亏格多项式.此处的结果形式更为简单.     

关于循环图的曲面嵌入

该文集中探讨循环图的曲面嵌入性质.决定了所有循环图的最小亏格(其中包括可定向亏格与不可定向亏格)和最大亏格.对于固定的整数l(≥3)和充分大的 自然数n,只有一种方式将4 -正则循环图C(n,l)嵌入到环面上使得其每一个面都是4 -边形.特别地,循环图$C(2l+2,l)$在加入若干条新边后可以同时将环面与Klein瓶进行三角剖分.     

关于图的上可嵌入性的一个注记

任韩和李刚在图的最大亏格综述一文"Survey of maximum genus of graphs" [J East China NormUniv Natur Sci, Sep. 2010, No. 5, 1-13] 中,全面地阐述了近30 年来关于图的最大亏格及其相关问题所取得的进展,并提出了如下两个猜想:
猜想1 设G 为简单连通图, 且G 的每条边含在一个三角形K₃ 中, 则G 是上可嵌入的.
猜想2 设c 为任意的正数, 则存在一个自然数N(c), 使得对每一个图G, 若G 的点数n ≥ N(c), 且最小度δ(G) ≥ cn, 则G 是上可嵌入的.
本文的主要工作是否定上述两个猜想, 同时探讨上述猜想成立的条件且得了一些新结果, 并提出有关进一步研究的问题.     

局部大边宽嵌入图的面圈和C-桥的结构

In this paper,we show that for a locally LEW-embedded 3-connected graph G in orientable surface,the following results hold:1) Each of such embeddings is minimum genus embedding;2) The facial cycles are precisely the induced nonseparating cycles which implies the uniqueness of such embeddings;3) Every overlap graph O(G,C) is a bipartite graph and G has only one C-bridge H such that CUH is nonplanar provided C is a contractible cycle shorter than every noncontractible cycle containing an edge of C.This ext...     

Bounding the Size of Equimatchable Graphs of Fixed Genus

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings.      

STRONG EMBEDDINGS OF PLANAR GRAPHS ON HIGHER SURFACES

In this paper, the authors discuss the upper bound for the genus of strong embeddings for 3-connected planar graphs on higher surfaces. It is shown that the problem of determining the upper bound for the strong embedding of 3-connected planar near-triangulations on higher non-orientable surfaces is NP-hard. As a corollary, a theorem of Richter, Seymour and Siran about the strong embedding of 3-connected planar graphs is generalized to orientable surface.     

Generalized Embedding Schemes

Covering projections are used to provide a logical foundation for the generalized embedding schemes which describe embeddings of graphs on closed surfaces that are not necessarily orientable. Criteria are provided to resolve the question of when such schemes describe the same embedding, and when that embedding is orientable. The schemes are then applied to derive some interpolation-type theorems for embeddings on closed surfaces.     

关于嵌入图的最大亏格

不依赖图的其它参数, 而主要依据图嵌入在定向曲面上的有关嵌入性质, 该文研究图的最大亏格.     

Genus embeddings of a type of graph

In 2003, Liu set up the joint tree model such that an embedding of a graph on a surface can be represented by a joint tree, further by an associated surface of it. In this paper, by dividing an associated surface into segments layer by layer and doing a sequence of operations on the layer division, we obtain the genus of complete bipartite graphs with a different method as before. Based on it, firstly, the genus of a new type of graph is derived. Then the number of non-homeomorphic genus embeddings of some complete bipartite graph is easily estimated.     

AT MOST SINGLE-BEND EMBEDDINGS OF CUBIC GRAPHS

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Automorphism properties of embedded graphs

This paper presents some conditions under which every automorphism of a graph, embedded on a surface of positive genus, maps face boundaries of the embedding to face boundaries. These results extend a result of Whitney which provides an analogous, but simpler, condition for planar graphs. The new result is applied to graphs on the torus to obtain a classification of many toroidal vertex-, edge-, and face-transitive graphs.     

可定向曲面上具有较短不可收缩圈图嵌入的唯一性

拓扑图论中的一个基本问题就是要决定图在一个(可定向)曲面上的嵌入之数目(既嵌入的柔性问题).H.Whitney的经典结果表明:一个3-连通图至多有一个平面嵌入;C.Thomassen的LEW-嵌入(大边宽度)理论将这一结果推广到一般的可定向曲面.本文给出了几个关于一般可定向曲面上嵌入图的唯一性定理.结果表明:一些具有大的面迹的可定向嵌入仍然具有唯一性.这在本质上推广了C.Thomassen在LEW-嵌入方面的工作.