完全多部图K_(1*r,3*(k-2))的m数

如果对一个图G的每个顶点v,任给一个k-列表L(v),使得G要么没有正常列表染色,要么至少有两种正常列表染色,则称图G具有M(k)性质.定义图G的m数为使得图G具有M(k)性质的最小整数k,记为m(G).已有研究表明,当k=3,4时,图K_(1*r,3*(k-2))具有M(k)性质,且当r≥2时,m(K_(1*r,3*(k-2)))=k.本文将上述结论推广到每一个k,证明了对任意r∈N~+,k≥3,图K_(1*r,3*(k-2))具有M(k)性质,且当k≥4,r≥(k-2)时,m(K_(1*r,3*(k-2)))=k.此外,得到图K_(1,3,3,3)的m数为4,该图是图K_(1*r,3*(k-2))中r=1,k=5时的特殊情况,同时也是现有研究中尚未解决的一个问题.     

若干完全三部图的色唯一性

用P(G,λ)表示图G的色多项式.若对任意图H,当P(H,λ)=P(G,λ)时都有H和G同构,则称图G是色唯一的.给出了以下结果:m≥2且k≥0时,完全三部图K(m,m,m+k)是色唯一的;m≥2且m+1>k≥0时,完全三部图K(m,m+1,m+k)是色唯一的.     

最大亏格、点度和围长

用g(G)和δ(G)分别表示一个图G的围长和顶点最小度. ζ(G)为图G的Betii亏数,主要证明了以下2个结果1)设G为k-边连通简单图,若对G中任意圈C,存在点x∈C满足dG(x)＞|V(G)|/(k-1)2+2)+k-g(G)+2,k=1,2,3,则G是上可嵌入的.且不等式的下界是最好的;2)设G为k-边连通简单图,则ζ(G)≤{max{1,m},k=1,max{1,1/(k-1)m -1}K=2,3 其中m= |V(G)|g(G)-6/g(G)2+(δ(G)-2)g(G)-4'且不等式的上界是可达的.进而得到了最大亏格一个比较好的下界.     

整数距离图G(D_(m,k,2))的点荫度

图G的点荫度va(G)是顶点集合V(G)能划分成的这样一些子集的最少数目,其中任一子集的点导出子图都是森林.整数距离图G(D)以全体整数作为顶点集,顶点u,v相邻当且仅当|u-v|∈D,其中D是一个正整数集.对于m2k≥2,令D_(m,k,2)=[1,m]\{k,2k}.该文得出了整数距离图G(D_(m,k,2))的点荫度的几个上、下界;进而,对于m≥4,有va(G(D_(m,1,2)))=[(m+4)/5];对于m=10q+j,j=0,1,2,3,5,6,有va(G(D_(m,2,2)))=[(m+1)/5]+1.     

与任意图正交的［0，ki］m1-因子分解

设G是一个图, k1,…, km是正整数.若图G的边能分解成m个边不交的［0,k1］-因子 F1,…,［0,km］-因子Fm,则称＝F1,…,Fm是G的一个［0,ki］m1-因子分解.如果H是G的一个有m条边的子图且对任意的1≤I≤m有｜E(H)∩E(Fi)｜＝1,则称与H正交.证明了若G是一个［0,k1＋…＋km－m＋1］-图,H是G的一个有m条边的子图,则图G有一个［0,ki］m1-因子分解与H正交.     

广义图K(n,m)的点强全色数

图G(V,E)的一个正常k-全染色σ称为G(V,E)的一个k-点强全染色,当且仅当v∈V(G),N[v]中的元素着不同颜色,其中N[v]={u vu∈V(G)}∪{v};并且χvTs(G)=m in{k存在G的一个k-点强全染色}称为G的点强全色数.本文确定了完全图Kn的广义图K(n,m)和乘积图Lm×Kn的点强全色数.     

关于图的Edmonds曲面对偶定理的一个注记

这个注记提供了如下两点:1.给出了Edmonds定理的一个简单而且是构造性的证明。2.表明这个定理与确定图的不可定向最大亏格的定理,即树除外,任何连通图G的不可定向最大亏格为     

图的指数多个最大亏格嵌人与完全图的亏格嵌入

本文研究一般图的最大亏格嵌入的计数问题及其应用.结果表明:一个连通图往往有指数级别多个最大亏格嵌入.特别地,一个简单的n阶3-正则图G至少具有(2~(1/2))~(m n (α/2))个不同的最大亏格潜入,其中α与m分别是G的最优树T的内部节点数目和G-T的奇连通分支数目.值得注意的是:(不同)图的最大亏格与最小亏格之间存在着某些必然联系.事实上,作为以上结果的一个直接应用,证明了如下结果:对于充分大的形如12s 4,12s 7,12s 10的自然数n,完全图K_n至少具有C2~(n/4)个不同的最小亏格嵌入,C是一个与n关于模12剩余类有关的常数.这些结果从本质上改进了V.P.Korzhik与H.-J.Voss所得到的结果,并且所用的方法更加直接而简洁.     

边剖分、点扩张与图的最大亏格的可约性

设γM（G）是连通图G=（V，E）的最大亏格，记EM^-（G）={e∈E(G)|G\e连通，且γM（G\e）=γM（G）}。若EM^-(G)≠0，则称G是γ（G）-可约的；否则称G是γM（G）-不可约的。本文证明了边的剖分不改变图的最大亏格可约性，点的扩张不改变上可嵌入图的最大亏格可约性；并给出了两类满足EM^-（G）=E（G）的非4-边连通图。     

高校应用数学学报第15卷(2000年)B辑(英文版)第3期目次和提要

3-边连通非简单图的最大亏格的一个注记黄元秋 (湖南师范大学数学系 )设 G为 3-边连通图 (不排除重边或环 ) ,且设γM( G)和β( G)分别为 G的最大亏格和 Betti数 .本文证明了γM( G)≥13β( G) ,从而回答了 Chen,Archdeacon及 Gross在 1996年所提出的一个问题 .整图的构造王力工　李学良　张胜贵 (西北工业大学应用数学系 )用两种新的方法给出了一些整图新类 ,也证明了寻找此类整图的问题与求解不定方程的问题是等价的 .其中一些整图类是具有无穷多个的 .这些图的发现是对寻找此类整图的一个新的贡献 .一类超二次二阶 Hamilton系统的…     

Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

The number of K m,m -free graphs

A graph is called H-free if it contains no copy of H. Denote by f _n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f _n (H) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and R?dl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log₂ f _n (H) is not known. We prove that f _n (K _m,m ) ≤ 2 ^O (n ^2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K _m,m ) is conjectured to be Θ(n ^2−1/m ). Our method also yields a bound on the number of K _m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K _3,3-free graphs of order n have more than 1/20·ex(n,K _3,3) edges.     

The characterization of totally geodesic submanifolds ofS m andCP m

We prove a theorem on ruled surfaces that generalizes a theorem of Ferus on totally geodesic foliations. On the basis of this theorem we obtain criteria for totally geodesic submanifolds ofS ^m andCP ^m that generalize and complement certain results of Borisenko, Ferus, and Abe. We give an application to the geodesic differential forms defined by Dombrowski in the case of submanifolds ofS ^m andCP ^m.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 106–116.The author is grateful to V. A. Toponogov for posing this problem and for attention to the work and to A. A. Borisenko for helpful criticisms.     

New Kloosterman sums identities over F2m for all m

Recently, Shin and Sung found new identities for Kloosterman sums over F_2^m with odd m. They posed the question whether similar results could be obtained for even m. In this paper, we will give a positive answer to this question. We will present new results that hold for any m and include as special cases the results of Shin and Sung in the case where m is odd.     

On Remez-type inequalities for polynomials in R m and C m

Remez-type inequalities provide estimates for the size of polynomials on given sets KR ^m (or C ^m ) when the magnitude of polynomials on largeldquo subsets of K is known. We shall study this question on smooth sets K in R ^m and C ^m and show how the smoothness of K effects the estimates.     

2m-Cycle Systems of K 2m+1\C m

For all m ≥ 3 the edges of complete graph on 2m + 1 vertices can he partitioned into m 2m-cycles and an m-cycle.