γ-可图序列与γ-完全子图

一个r-图是一个无环的无向图,其中任何两个顶点之间至多被r条边连接.一个m+1个顶点的r-完全图,记为K_(m+1)^((r)),是一个m+1个顶点的r-图,其中任何两个顶点之间恰好被r条边连接.一个非增的非负整数序列π=(d_1,d_2,…,d_n)称为是r-可图的如果它是某个n个顶点的r-图的度序列.一个r-可图序列π称为是蕴含(强迫)K_(m+1)((r)),是一个m+1个顶点的r-图,其中任何两个顶点之间恰好被r条边连接.一个非增的非负整数序列π=(d_1,d_2,…,d_n)称为是r-可图的如果它是某个n个顶点的r-图的度序列.一个r-可图序列π称为是蕴含(强迫)K_(m+1)^((r))可图的如果π有一个实现包含K_(m+1)((r))可图的如果π有一个实现包含K_(m+1)^((r))作为子图(π的每一个实现包含K_(m+1)((r))作为子图(π的每一个实现包含K_(m+1)^((r))作为子图).设σ(K_(m+1)((r))作为子图).设σ(K_(m+1)^((r)),n)(τ(K_(m+1)((r)),n)(τ(K_(m+1)^((r)),n))表示最小的偶整数t,使得每一个r-可图序列π=(d_1,d_2,…,d_n)具有∑_(i=1)((r)),n))表示最小的偶整数t,使得每一个r-可图序列π=(d_1,d_2,…,d_n)具有∑_(i=1)ⁿ d_i≥t是蕴含(强迫)K_(m+1)n d_i≥t是蕴含(强迫)K_(m+1)^((r))-可图的.易见,σ(K_(m+1)((r))-可图的.易见,σ(K_(m+1)^((r)),n)是Erds等人的一个猜想从1-图到r-图的扩充且τ(K_(m+1)((r)),n)是Erds等人的一个猜想从1-图到r-图的扩充且τ(K_(m+1)^((r)),n)是经典Turan定理从1-图到r-图的扩充.本文给出了蕴含K_(m+1)((r)),n)是经典Turan定理从1-图到r-图的扩充.本文给出了蕴含K_(m+1)^((r))的r-可图序列的两个简单充分条件.此两个条件包含了Yin和Li在[Discrete Math.,2005,301:218-227]中的两个主要结果和当n≥max{m((r))的r-可图序列的两个简单充分条件.此两个条件包含了Yin和Li在[Discrete Math.,2005,301:218-227]中的两个主要结果和当n≥max{m²+3m+1-[(m2+3m+1-[(m²+m)/r],2m+1+[m/r]]}时,σ(K_(m+1)2+m)/r],2m+1+[m/r]]}时,σ(K_(m+1)^((r)),n)之值.此外,我们还确定了当n≥m+1时,τ(K_(m+1)((r)),n)之值.此外,我们还确定了当n≥m+1时,τ(K_(m+1)^((r)),n)之值.     

蕴含K6-Z6-可图序列的刻划

对于给定的图日,若可图序列π有一个实现G以H为其子图,则称π为蕴含日一可图的．在本文,作者刻划了蕴含K6-Z6-可图序列．     

蕴含K4＋P2-可图序列的刻划

对于给定的图H,若存在可图序列π的一个实现包含H作为子图,则称π为蕴含H-可图的．Gould等人考虑了下述极值问题的变形：确定最小的偶整数σ（H,n）,使得每个满足σ（π）≥σ（H,n）的n项可图序列π=（d1,d2,…,dn）是蕴含H-可图的,其中σ（π）=∑di．本文刻划了蕴含K4＋P2-可图序列,其中K4＋P2是向致的一个顶点添加两条悬挂边后构成的简单图．这一刻划导出σ（K4＋P2,n）的值．     

蕴含W6-可图序列

摘要对于给定的图日,如果可图序列π有一个实现包含日作为子图,则称丌是蕴含H-可图的．本文给出了可图序列π蕴含W6-可图的一个充分条件,其中Wτ是τ个顶点的轮图．     

关于图的余树的奇连通分支数的内插定理

本文研究了连通图的余树的奇连通分支数与其可定向嵌入的关系.我们先给出了关于连通图的余树的奇连通分支数的内插定理.作为其应用,我们推广了Xuong和刘彦佩关于图的最大亏格的计算公式,并且证明了如下结果:任意一个连通图G一定满足下列条件之一: (a)对于任意的满足γ(G)≤g≤γM(G)整数g,只要图G嵌入到可定向曲面Sg上,就存在支撑树T,使g-1/2β(G)-ω(T)),其中,γ(G)与γM(G)分别是图G的最小和最大亏格,β(G)与ω(T)分别是图G的Betti数和由T确定的余树的奇连通分支数; (b)对连通图G的任意一个支撑树T,G可以嵌入某个可定向曲面上使其恰好有ω(T) 1个面.特别地,我们给出了所有非平面的3-正则的Hamilton图G所嵌入的可定向曲面的亏格的计算公式.     

给定团数的连通图的最小代数连通度

图G的拉普拉斯矩阵的第二小特征值称为图G的代数连通度.在给定团数ω的n阶连通图中,本文刻画了具有最小代数连通度的图为风筝图PK_(n-ω,ω),其中风筝图PK_(n-ω,ω)是由完全图K_ω在某一点上引出一条悬挂路P_(n-ω)而得到的图.同时,对风筝图PK_(n-ω,ω)的代数连通度的一些性质也做了讨论.     

几类图弱控制的广义束缚数

对任一图G,其弱控制的束缚数,广义束缚数分别定义为bw(G)=min{|E‖E(∪)E(G),且γw(G-E)＞γω(G)}.b'ω(G)=min{t|(A)E(∪)E(G),如果|E|=t,则有γω(G-E)＞γw(G)}.在本文中我们给出了几类图的弱控制的广义束缚数的精确值,称b'ω(G)=1图为弱控制去边临界图,并研究了正则图是弱控制去边临界图的充要条件,以及一般图和树的必要条件.     

匹配数与控制数相等的图的结构性质

我们分别用γ(G)，β(G)和α(G)表示图G的控制数、匹配数和覆盖数，对任意连通图，有γ(G)≤β(G)≤α(G)成立，1998年，Randerath和Volkmann给出了控制数等于覆盖数的图的特征，本文首先证明了匹配数与控制数相等的图其最小度不超过2，而后给出了最小度为2的图的结构性质。     

蕴含W_6-可图序列

对于给定的图H,如果可图序列π有一个实现包含H作为子图,则称π是蕴含H-可图的.本文给出了可图序列π蕴含W_6-可图的一个充分条件,其中W_r是r个顶点的轮图.     

c-圈图的Laplace谱半径和最大度

设G是一个n阶的简单连通图,符号(d_1,d_2,...,d_n)表示G的度序列,其中d_1≥d_2≥···≥d_n,用符号?(G)表示G的最大度,而符号λ(G)表示G的Laplace谱半径.一个c-圈图是一个恰有n+c-1条边的n阶简单连通图,而符号C(n,?;c)表示最大度等于?的所有n阶c-圈图的集合.本文确定了当0≤c≤1/2(?-1)(?-2)时,C(n,?;c)中所有取得最小Laplace谱半径的极图,并分别确定了当?≥[n+2/3]且d_4≥2或?≥[n/3]+1且d_4=1时,C(n,?;1)中唯一取得最大Laplace谱半径的极图.进一步地,还证明了对于两个n阶的单圈图G和G′,如果?(G)≥[11n/30]+2且?(G)?(G′),则λ(G)λ(G′),并且界"[11n/30]+2"是最佳的.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r, n) such that every n-term graphic sequence π = (d1, d2,..., dn) with term sum σ(π) = d1 + d2 +…+ dn ≥σ(Kr,r, n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. πr has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined.     

The signless Laplacian spectral radius of graphs with given degree sequences

In this paper, we investigate the properties of the largest signless Laplacian spectral radius in the set of all simple connected graphs with a given degree sequence. These results are used to characterize the unicyclic graphs that have the largest signless Laplacian spectral radius for a given unicyclic graphic degree sequence. Moreover, all extremal unicyclic graphs having the largest signless Laplacian spectral radius are obtained in the sets of all unicyclic graphs of order n with a specified number of leaves or maximum degree or independence number or matching number.     

阶数极大的临界全控制图

称一个没有孤立点的图G 为临界全控制图, 如果G 满足对于任何一个不与悬挂点相邻的顶点v, G - v 的全控制数都小于G 的全控制数. 如果G 的全控制数记为γ_t, 则称这样的临界全控制图G 为γ_t- 临界的. 如果G 是γ_t- 临界的, 且阶数为n, 则n ≤ Δ(G)(γ_t(G)- 1) + 1, 其中Δ(G) 是G 的最大度. 本文将证明对γ_t = 3, 这个阶数的上界是紧的, 并给出所有满足n = Δ(G)(γ_t(G)- 1) + 1 的3-γ_t- 临界图.     

A Construction of Sequentially Cohen-Macaulay Graphs

For every simple graph G,a class of multiple clique cluster-whiskered graphs Geπm is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex IndGeπm is sequentially Cohen-Macaulay.The properties of the graphs Geπm and Gπ constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind Gπ and the calculation of Betti numbers of the cover ideal Ic(Geπ").We also prove that the complex △ =IndH is strongly shellable and pure for either a Boolean graph H =Bn or the full clique-whiskered graph H =Gw of G,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△) and the cover ideal Ic(H) have linear quotients.     

线图上的团染色问题

设G=(V,E)为简单图, G的每个至少有两个顶点的极大完全子图称为G的一个团. 图的团染色定义为给图的点进行染色使得图中没有单一颜色的团, 也就是说每一个团具有至少2种颜色.图的一个k-团染色 是指用k 种颜色给图的点着色使得图G 的每一个团至少有2种颜色.图G的团染色数\chi_{C}(G)是指最小的数k使得图G 存在k-团染色. 首先指出了完全图的线图的团染色数与推广的Ramsey 数之间的一个联系, 其次对于最大度不超过7的线图给出了一个最优团染色的多项式时间算法.     

Chromatic number and complete graph substructures for degree sequences

Given a graphic degree sequence D, let χ(D) (respectively ω(D), h(D), and H(D)) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is D. It is proved that χ(D)≤h(D). Moreover, it is shown that a subdivision of a clique of order χ(D) exists where each edge is subdivided at most once and the set of all subdivided edges forms a collection of disjoint stars. This bound is an analogue of the Hajós Conjecture for degree sequences and, in particular, settles a conjecture of Neil Robertson that degree sequences satisfy the bound χ(D) ≤ H(D) (which is related to the Hadwiger Conjecture). It is also proved that χ(D) ≤ 6/5 ω(D)+ 3/5 and that χ(D) ≤ 4/5 ω(D) + 1/5 Δ(D)+1, where Δ(D) denotes the maximum degree in D. The latter inequality is related to a conjecture of Bruce Reed bounding the chromatic number by a convex combination of the clique number and the maximum degree. All derived inequalities are best possible     

On Domination and Independence Numbers of Graphs

A set S of vertices of a graph G is dominating if each vertex x not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number, γ(G), is the order of the smallest dominating set in G. The independence number, α(G), is the order of the largest independent set in G. In this paper we characterize bipartite graphs and block graphs G for which γ(G) = α(G).     

可图序列偏序集的极小元

ｎ项非增非负整数序列是可图的,若是某个阶简单图的度序列．所有项和为２ｍ、迹为ｆ的ｎ项可图序列的集合Ｇ_ｎ,ｍ,ｆ在优超关系下是一个偏序集．本文刻划了偏序集Ｇ_ｎ,ｍ,ｆ的极小元,并确定各种可图序列偏序集中极小元的个数．     

Radius two trees specify χ-bounded classes

A class of graphs χ is said to be χ-bounded, with χ-binding function f, if for all G ? Γ, χ (G) ≦ f (ω(G)), where χ(G) is the chromatic number of G and ω(G) is the clique number of G. It has been conjectured that for every tree T, the class of graphs that do not induce T is χ-bounded. We show that this is true in the case where T is a tree of radius two.