双圈覆盖问题

Seyntour[1]与Szekeres[5]猜想，每一个无割边的图G具有一个圈集合 使G中的每个边存在于 的两个圈中。本证明此猜想成立当且仅当它对没有非平凡的三个边割的图成立。     

On the hamiltonicity of line graphs of locally finite, 6‐edge‐connected graphs

The topological approach to the study of infinite graphs of Diestel and KÜhn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4‐edge‐connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6‐edge‐connected graph with a finite number of ends, each of which is thin, is hamiltonian.     

全矩阵环上理想包含图的自同构群

设R是一个环,其上的理想包含图,记为Γ_I(R),是一个有向图,它以R的非平凡左理想为顶点,从R的左理想I_1到I_2有一条有向边当且仅当I_1真包含于I_2.环R上的理想关系图,记为Γ_i(R),也是一个有向图,它以R为顶点集,从R中元素A到B有一条有向边当且仅当A生成的左理想真包含于B生成的左理想.设F_q为有限域,其上n阶全矩阵环记为M_n(F_q),本文刻画了环M_n(F_q)上的理想包含图以及理想关系图的任意自同构.     

Graph homomorphisms through random walks

In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff–Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge‐transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides a proof of the fact that the Coxeter graph is a core. Also, we obtain some information about the cores of vertex‐transitive graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 15–38, 2003     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

A contribution to queens graphs: A substitution method

A graph G is a queens graph if the vertices of G can be mapped to queens on the chessboard such that two vertices are adjacent if and only if the corresponding queens attack each other, i.e. they are in horizontal, vertical or diagonal position.We prove a conjecture of Beineke, Broere and Henning that the Cartesian product of an odd cycle and a path is a queens graph. We show that the same does not hold for two odd cycles. The representation of the Cartesian product of an odd cycle and an even cycle remains an open problem.We also prove constructively that any finite subgraph of the rectangular grid or the hexagonal grid is a queens graph.Using a small computer search we solve another conjecture of the authors mentioned above, saying that K_3,4 minus an edge is a minimal non-queens graph.     

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...     

Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.     

Ideals and graphs,Gröbner bases and decision procedures in graphs

The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs, minimal vertex covers, cliques, edge covers and matchings with algebraic tools. All such procedures are implemented in Maple 9.5.     

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.     

没有K4-图子式的图的无圈边色数

一个图G 的无圈k- 边染色是指G 的一个正常的不产生双色圈的k- 边染色. G 的无圈边色数a′(G) 定义为使得G 有一个无圈k- 边染色的最小的整数k. 本文完全刻画了最大度不为4 的没有K₄-图子式的图的无圈边色数.     

Disjoint Hamilton cycles in the random geometric graph

We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge‐length. For fixed k?1, weprove that the first edge in the process that creates a k‐connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge‐disjoint Hamilton cycles (for even k), or (k?1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge‐disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2‐connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the ?_p norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299‐322, 2011     

一类混合图的结构及其特征空间

主要讨论具有如下性质的一类连通混合图G:其所有非奇异圈恰有一条公共边,且除了该公共边的端点外,任意两个非奇异圈没有其它交点.本文给出了图G的结构性质,建立了其最小特征值λ1(G)(以及相对应的特征向量)与某个简单图的代数连通度(以及Fiedler向量)之间联系,并应用上述联系证明了λ1(■)≤α(G),其中G是由G通过对其所有无向边定向而获得,α(■)为■的代数连通度.     

On removable cycles through every edge

Mader and Jackson independently proved that every 2‐connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G/E(C) is 2‐connected. This paper considers the problem of determining when every edge of a 2‐connected graph G, simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from G remains 2‐connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge‐disjoint removable cycles such that each contains one of the distinguished edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 155–164, 2003     

关于超欧拉图的一个注记

设G是无向无环的有限图 ,若G有一个生成子图是欧拉图 (Euler) ,则称G是超欧拉图 (Supereulerian) .本文不利用收缩方法 ,直接证明了 :当图G至多差一边有两棵边不相交的生成树时 ,G是超欧拉图或者G有割边 .     

若干圈限制平面图的星边染色

图G的星边染色是指G的一个正常边染色,使得G中任一长为4的路和长为4的圈均不是2-边染色的.图G的星边色数X's(G)表示图G有星边染色的最小颜色数.设G是最大度为△的平面图,我们证明了:(1)若G不含4-圈,则X'st(G)≤「1.5△」+15;(2)若g≥5,则X'st(G)≤「1.5△」+10;(3)若 g=7,...     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

Search games with immobile hider

In this paper we discuss a type of search game with immobile hider on a graph. If the graph is weakly cyclic (i.e. no edge lies on two different cycles) we determine the value of the game and give an easy way to compute optimal strategies for the players. The value is half the length of a quasi-Euler path, i.e. a closed path of minimal length that visits all points of the network.     

Regularity,depth and arithmetic rank of bipartite edge ideals

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.