两个图的积的S—不可收缩性

本文从组合角度明确给出了两个图的积是Ｓ－不可收缩的特征。     

二部图的E-H-不可收缩性

By End（G） and hEnd（G） we denote the set of endomorphisms and half-strong endomorphisms of a graph G respectively. A graph G is said to be E-H-unretractive if End（G） = hEnd（G）. A general characterization of an E-H-unretractive graph seems to be difficult. In this paper, bipartite graphs with E-H-unretractivity are characterized explicitly.     

一致可积函数的非标准刻画

设(X,A,μ)是内有限可加测度空间.首先给出了S-可积的等价条件,进而给出了一致可积函数的非标准刻画,即可测函数集{fi)i∈I是一致可积的当且仅当对于任意的i∈*I,fi都是S-可积的.     

两个新的可积类型的微分方程

研究方程y^1=f（x,y）的可积性,建立了两个可积类型的微分方程y^1=g（x）h（^y——φ（x））＋^φ^1（x）——φ（x）y和^1=^φ（x）——xg（^y——φ（x）＋x^8（^y——φ（x））＋^φ^1（x）——φ（x））y应用变换y=φ（x）u,它们可分别化为x^au′=g（x）h（u）和^dx——du=g（u）x＋h（u）x^β进行求解．     

两个可积系统及其约化

该文引入了一个李代数,然后定义了其相应的两个圈代数,利用圈代数构造了两个等谱问题,其相容性条件导出了两个可积动力系统.通过约化这样的系统,得到了某些有趣的非线性方程,如Burgers方程、组合KdV-MKdV方程和Kuramoto-Sivashinsky方程以及KdV方程的一种推广形式.最后,利用贝尔多项式讨论了广义KdV方程的可积性质,包括双线性形式、Lax对、贝克隆变换和无穷守恒律等.     

两类(2+1)维可积系统

利用一个广义等谱问题的相容性得到了一个广义零曲率方程.作为其应用,首先利用loop代数设计了两个广义等谱问题,然后利用零曲率方程导出了两类(2+1)维Lax可积系统.     

6连通图中的可收缩边

Kriesell(2001年)猜想：如果κ连通图中任意两个相邻顶点的度的和至少是2[5κ/4]-1则图中有κ-可收缩边．本文证明每一个收缩临界6连通图中有两个相邻的度为6的顶点，由此推出该猜想对κ=6成立。     

对称扩散过程的对数Sobolev不等式、指数可积性和熵的指数衰减性(英)

本文讨论了对称扩散过程的指数可积性，熵的指数衰减性与Sobolev不等式之间的等价关系．并给出了对称扩散过程的对数Sobolev不等式成立的一个充分条件．     

保向形式Jacobi行列式的可积性

考虑各向异性Sobolev类上的保向形式的Jacobi行列式的可积性, 给出了保证Jacobi行列式可积的一个充分条件, 这可看作是Iwaniec和Sbordone结果的优化.     

两类积图的邻点可区别全染色

本文.证明了,当n≥2时,Xat(K_n×K′_n)=2n;当p,q≥2时,Xat(C_(2p)×K_(2q))=2q 3,其中K_n×K′_n是两个不同标号完全图的积图,C_(2p)×K_(2q)是偶圈和偶阶完全图的积图.     

Characterization of Product Anti-Magic Graphs of Large Order

A graph G is product anti-magic if one can bijectively label its edges with integers 1, . . . ,e(G) so that no two vertices have the same product of incident labels. This property was introduced by Figueroa-Centeno, Ichishima, and Muntaner-Batle who in particular conjectured that every connected graph with at least 4 vertices is product anti-magic. Here, we completely describe all product anti-magic graphs of sufficiently large order, confirming the above conjecture in this case. Our proof uses probabilistic methods. Reverts to public domain 28 years from publication. Partially supported by the National Science Foundation, Grant DMS-0457512.     

图的强乘积的带宽

本文讨论了由两个图的强乘积所导出的一些特殊图的带宽.     

双线性型图与交错型图不是完美图

本文证明了双线性型图与交错型图都不是完美图,从而解决了双线性型图与交错型图的完美图判别问题.     

两类新的匹配唯一性图族

本文研究了具有度序列(13,2S-4,3)的图的匹配唯一性,给出了T(1,4,n)∪(s∪i=0Cpi)(n 4)与T(1,5,n)∪(s∪i=0Cpi)(n 5)及其补图匹配唯一的充要条件.     

关于乘积图的二维带宽问题

给出一般乘积图的二维带宽的界，并解决一类乘积图的二维带宽问题．最后给出完全ｋ部图的二维带宽。     

Thickness-Two Graphs Part Two: More New Nine-Critical Graphs,Independence Ratio,Cloned Planar Graphs,and Singly and Doubly Outerplanar Graphs

There are numerous means for measuring the closeness to planarity of a graph such as crossing number, splitting number, and a variety of thickness parameters. We focus on the classical concept of the thickness of a graph, and we add to earlier work in [4]. In particular, we offer new 9-critical thickness-two graphs on 17, 25, and 33 vertices, all of which provide counterexamples to a conjecture on independence ratio of Albertson; we investigate three classes of graphs, namely singly and doubly outerplanar graphs, and cloned planar graphs. We give a sharp upper bound for the largest chromatic number of the cloned planar graphs, and we give upper and lower bounds for the largest chromatic number of the former two classes.     

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K _n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product of graphs. As the main result of this paper, we prove that for any two graphs G ₁ and G ₂ with η(G ₁) = h and η(G ₂) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following:

两个完全图的乘积的树宽

本文确定了乘积图Km×Kn的树宽.我们的结果是若m和n都是偶数,且m≥n,或m是奇数而n是偶数,或m和n都是奇数且n≥m,则Km×Kn的树宽是TW(Km×Kn)=n(m+1)/2-1.这恰好是图Km×Kn的带宽.     

Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds

The author introduces the w-function defined on the considered spacelike graph M.Under the growth conditions w = o(log z) and w = o(r),two Bernstein type theorems for M in Rmn+ mare got,where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively.As the ambient space is a curved pseudoRiemannian product of two Riemannian manifolds(Σ1,g1) and(Σ2,g2) of dimensions n and m,a Bernstein type result for n = 2 under some curvature conditions on Σ1 and Σ2 and the growth condition w = o(r) is also got.As more general cases,under some curvature conditions on the ambient space and the growth condition w = o(r) or w = o(√r),the author concludes that if M has parallel mean curvature,then M is maximal.