与直径和围长有关的最大亏格的下界

本文证明了如下结果:设G为直径为d的简单图,若G的围长不小于d,则当d为不小于4的偶数时,有ξ(G)≤1,即G是上可嵌入的;当d为不小于3的奇数时,有ξ(G)≤2,即γM(G)≥1/2β(G)-1.     

与直径和围长有关的图的最大亏格

利用图的直径和围长来研究图的最大亏格的下界,得到了如下结果：设G是直径为d的简单图,若G的围长不小于d（其中d为不小于3的整数）,则ξ（G）≤2,即γM（G）≥1/2β（G）-1．而且,在这种意义下,所得到的界是最好的．     

一个关于余直径的度条件

本文研究关于余直径的度条件,主要结果是,若G是3-连通图,其顶点集为V(G)={v1,v2,…,vn),其度序列为: d(v1)≤d(v2)≤…≤d(vn)则对G的任意一条边e都存在特征点对va,vb,使G中过e的最长圈至少为d(va)+d(vb)-1.在相同的条件下我们有d*(G)≥ m-2,     

图的围长与无圈边色数之间的关系

对于一个图G的正常边着色，如果此种边着色使得该图没有2—色的圈，那么这种边着色被称为是G的无圈边着色．用d(G)表示图G的无圈边色数，即G的无圈边着色中所使用的最小颜色数．Alon N，Sadakov B and Zaks A在[1]中有如下结果：对于围长至少是2000△(G)log△(G)的图G，有d(G)≤△ 2，其中△是图G的最大度．我们改进了这个结果，得到了如下结论：对于围长至少是700△(G)log△(G)的图G，有d(G)≤△ 2．     

2—连通正则二部图的长圈

Dirac 定理指出:若 G 是 n 个顶点的2-连通图,(?){d(x)}≥k,则 G 有长至少为 min(2k,n)的圈(见[1]).‖本文把 Dirac 定理应用到2-连通正则二部图,得到如下的结果:定理1 设 G 是2-连通 k-正则二部图,G 的顶点数为 n,则 G 有长至少为 min(4k,n)的圈(k≥2).‖     

关于图的上可嵌入性的一个充分条件

本文主要证明:设G是一个(k+1)-边连通的n阶简单图,其围长为g,如果对G的任意独立集I(G)={v_i|1≤i≤k~2+2},k=0,1,2,均满足那么图G是上可嵌入的,而且下界是紧的.     

平面图的绑定数

图G的绑定数b(G)是指边集合的最少边数,当这个边集合从G中去掉后所 得图的控制数大于G的控制数． Fischermann等人在[3]中给出了两个猜想: (1)如果 G是一个连通的平面图且围长g(G)≥4,则b(G)≤5;(2)如果G是一个连通的平面图且 围长g(G)≥5,则b(G)≤4．设n3表示度为3的顶点个数,r4和r5分别表示长为4和 5的圈的个数．本文,我们证明了如果r4<(5n3)/2 10,则猜想1成立;如果r5<12,则猜 想2成立．     

哈密尔顿性和部分平方图的独立集

设G是一个图,G的部分平方图G*满足V(G*)=V(G),E(G*)=E(G)∪{uv:uv■E(G),且J(u,v)≠■},这里J(u,v)={w∈N(u)∩N(v):N(w)■N[u]∪N[v]}.利用插点方法,证明了如下结果:设G是k-连通图(k2),b是整数,0min {k,(2b-1+k)/2}(n(Y)-1),则G是哈密尔顿图.同时给出图是1-哈密尔顿的和哈密尔顿连通的相关结果.     

关于图的L(3,2,1)-标号问题

图 G的 L (2 ,1) -标号是一个从顶点集 V(G)到非负整数集的函数 f (x) ,使得若 d(x,y) =1,则 | f (x)- f (y) |≥ 2 :若 d(x ,y) =2 ,则 | f (x) - f (y) |≥ 1.图 G的 L (2 ,1) -标号数λ(G)是使得 G有 max{ f (v) :v∈ V(G) } =k的 L(2 ,1) -标号中的最小数 k.本文将 L(2 ,1) -标号问题推广到更一般的情形即 L(3,2 ,1) -标号问题 ,并得出了细分图、Descartes图的 λ3 (G)的上界 .     

第二大根小于1的简单图

设G为n阶简单图,λ2(G)为G的第二大特征根．我们给出了所有使λ2(G)<1 的偶图,以及使λ2(G)<1、围长不小于4的非偶图．     

子群的S-条件置换性对有限群结构的影响

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.     

有限群的素数幂阶S-拟正规嵌入子群

设G为有限p-可解群,其中p为|G|的奇素因子.若P为G的Sylow p-子群且最小生成系含d个元素.考虑集合M_d(P)={P_1,…,P_d},其中P_1,…,P_d是P的极大子群且满足(?)P_i=φ(P).证明了若M_d(P)中每个元在G中是S-拟正规嵌入的,则G为p-超可解群.作为应用,还得到了一些进一步的结论.     

关于图的3-条件色数

For integers k0,r0,a(k,r)-coloring of a graph G is a proper k-coloring of the vertices such that every vertex of degree d is adjacent to vertices with at least min{d,r}diferent colors.The r-hued chromatic number,denoted byχr(G),is the smallest integer k for which a graph G has a(k,r)-coloring.Define a graph G is r-normal,ifχr(G)=χ(G).In this paper,we present two sufcient conditions for a graph to be 3-normal,and the best upper bound of 3-hued chromatic number of a certain families of graphs.     

关于有限群的S-半置换子群

Let d be the smallest generator number of a finite p-group P and let Md（P） = {P1,...,Pd} be a set of maximal subgroups of P such that ∩di=1 Pi = Φ（P）. In this paper, we study the structure of a finite group G under the assumption that every member in Md（Gp） is S-semipermutable in G for each prime divisor p of ｜G｜ and a Sylow p-subgroup Gp of G.     

给定直径的单圈图的超Wiener指数

The hyper-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener index W W(G) is defined as WW(G) =1/2∑_(u,v)∈V(G)(d_G(u, v) + d_G~2(u,v)) with the summation going over all pairs of vertices in G, d_G(u,v) denotes the distance of the two vertices u and v in the graph G. In this paper,we study the minimum hyper-Wiener indices among all the unicyclic graph with n vertices and diameter d, and characterize the corresponding extremal graphs.     

有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群

Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p-subgroup of G. If d is the smallest generator number of P, then there exist maximal subgroups P1, P2,..., Pd of P, denoted by Md(P) = {P1,...,Pd}, such that di=1 Pi = Φ(P), the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md(P) is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.     

An Extension of the Win Theorem: Counting the Number of Maximum Independent Sets

Win proved a well-known result that the graph G of connectivity κ(G) withα(G) ≤κ(G) + k-1(k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the Win theorem in case when κ(G) = 1 to the following: Let G be a simple connected graph of order large enough such that α(G) ≤ k + 1(k ≥ 3) and such that the number of maximum independent sets of cardinality k + 1 is at most n-2k-2. Then G has a spanning k-ended tree.     

树和单圈图的斜谱矩

给定图$G$,对图$G$的每条边确定一个方向,称为$G$的定向图$G^\sigma$, $G$称为$G^\sigma$的基础图. $G^\sigma$的斜邻接矩阵$S(G^\sigma)$是反对称矩阵,其特征值是0或纯虚数. $S(G^\sigma)$所有特征值的$k$次幂之和称为$G^\sigma$的$k$阶斜谱矩,其中$k$是非负整数.斜谱矩序列可用于对图进行排序.本文主要研究定向树和定向单圈图的斜谱矩,并对这两类图的斜谱矩序列依照字典序进行排序.首先确定了直径为$d$的树作为基础图的所有定向树中,斜谱矩序最大的$2\lfloor\frac{d}{4}\rfloor$个图; 然后确定以围长为$g$的单圈图作为基础图的所有定向单圈图中, 斜谱矩序最大的$2\lfloor\frac{g}{4}\rfloor+1$个图.     

具有弱$s$-置换嵌入子群的有限群

Let P be a Sylow p-subgroup of a group G with the smallest generator number d,where p is a prime.Denote by M_d(P) = {P_1,P_(2,...,)P_d} a set of maximal subgroups of P such that φ(P) = ∩_(n=1)~dP_n.In this paper,we investigate the structure of a finite group G under the assumption that the maximal subgroups in M_d(P) are weakly s-permutably embedded in G,some interesting results are obtained which generalize some recent results.Finally,we give some further results in terms of weakly s-permutably embedded subgroups.     

嵌入曲面的特殊图的边染色

设图\,$G$\,是嵌入到欧拉示性数\,$\chi(\Sigma)\geq 0$\,的曲面\,$\Sigma$\,上的图, $\chi'(G)$\,和\,$\Delta(G)$\,分别表示图\,$G$\,的边色数和最大度. 如果\,$\Delta(G)\geq 4$\,且\,$G$\,满足以下条件: (1)\,图$G$中的任意两个三角形$T_1$, $T_2$的距离至少是$2$; (2)\,图\,$G$\,中\,$i$-圈和\,$j$-圈的距离至少是\,$1$, $i,j\in\{3,4\}$; (3)\,图\,$G$\,中没有\,$5$-圈, 则有\,$\Delta(G)=\chi'(G)$.