图的上可嵌入性的邻域条件

用NG（u）表示一个图G中任意点u的邻域集．本文主要证明了下述结果：设G是无环图,对G中任意相邻的点u和υ,即uυ∈E（G）,若如下两条件之一满足：（1）|NG(u)∩NG(υ)≥2;（2）G是2-点连通的图,且|NG(u)∩NG(υ)|≥1,则G是上可嵌入的．     

哈密顿线图的一个充分条件

对于图G的任意边e=uv,边的度定义为d(e)=d(u)+d(v),其中d(u)和d(v)分别为顶点u和v的度.本文的主要结果是: 设G是几乎无桥的p≥2阶简单连通图,且G(?)K_(1,p-1),若对任意相距为2的两边e_1和e_2,d(e_1)+d(e_2)≥2p-6,则G有一个D—闭迹,从而G的线图L(G)是哈密顿的.     

点度与图的上可嵌入性

本文证明了:(1) 设G是2-连通简单图,且不含K_3,若对任意一对距离为2的点u,u,有max{d(u),d(u)}>n/3-1,其中n=|V(G)|,则G是上可嵌入的,且条件中不等式的界"n/3-1"是不可达的;(2) 设G是3-连通简单图,若对任意依次相邻的三点u,u,W,有max{d(u),d(u),d(w)}≥n/6+1,其中n=|V(G)|,则G是上可嵌入的,且条件中不等式的界"n/6+1"是最好的.     

直径为4的树的奇强协调性

对简单图G=〈V,E〉,如果存在一个映射f:V→{0,1,2,…,2 E-1}满足1)对任意的u,v∈V,若u≠v,则f(u)≠f(v);2)对任意的e1,e2∈E,若e1≠e2,则g(e1)≠g(e2),此处g(e)=f(u)+f(v),e=uv;3){g(e)e∈E}={1,3,5,…,2 E-1},则称G为奇强协调图,f称为G的奇强协调标号.给出了直径为4的树的奇强协调标号.     

导出子图和周长的局部定理

设G是n阶2-连通图，3≤c≤n.本文绘出对于图G的每一个同构于K1.3或Z1的导出子图L，若d（u）且如果dL（u，v）=2有（v）=min{，|M3（u）|/2}这里M3（u）={v|dc（u,v）≤3}，则G包含长至少为c的圈．     

无爪图泛圈性的邻域并条件

证明了,若G是一个p-阶3-连通无爪图,p≠10,11,15,并对G中任意两个不相邻的点u和v,满足|N(u)∪N(v)|≥(p-1)/2,则G是泛圈图.     

若干图的点强全染色(英文)

对图G及正整数k，映射f：满足：（1）任意e1，e3，如果e1，e2是相邻或相关联的，则有；（2）对u，v，w（G）有，则称f为G的一个k-点强全染色，并且K|G的社点强全染色称为G的点强全色数．本文讨论了一些特殊困的点强全色数，并提出了一个猜想：若G为每一分图的阶数不小于6的图，则（G），其中（G）为本文中定义的一新参数．     

幂图的两个遗传性质

图G=(V,E)被称为点可迹的,如果对任意一点u,G中存在Hamilton链使u为其一端点;图G被称为{u}-Hamilton链连通的,如果对任意v∈V\u,G中存在Ha-milton链使u,v为其两端点。对于任意V_0V,0≤|V_0|≤h(或V_0V\u.0≤|V_0|≤h),如果G\V_0是点可迹的(或{u}-Hamilton连通的),则称G为h-点可迹的(或h-{u}-Hamilton连通的)。本文证明了:若G是h-点可迹的(或h-{u}-Hamilton连通的),则其幂图G~h是(h+2k-2)-点可迹的(或(h+2k-2)-{u}-Hamilton连通的)(|V|≥h+2k+1)。     

关于P_(r,(2s-1))的奇优美标号

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2 |E|-1}满足1)对任意的u,v∈V,若u≠v,则(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4){g(e)|e∈E}={1,3,5,…,2|E|-1},则称G是奇优美图,f称为G的奇优美标号.Gnanajoethi提出了一个猜想:每棵树都是奇优美的.证明了图P_(r,(2s-1)是奇优美图.     

联图Fn∨Pm的邻点可区别全染色

设G(V,E)是阶数至少为2的简单连通图,k是正整数,V∪E到{1,2,3,…k}的映射f满足:对任意uv,uw∈E(G),u≠w,有f(uv)≠f(vw);对任意uv∈E(G),有f(u)≠f(v), f(u)≠f(uv),f(v)≠f(uv);那么称f为G的k-正常全染色,若f还满足对任意uv∈E(G),有G(u)≠C(v),其中C(u)={f(u)}∪{f(uv)|uv∈E(G),v∈V(G)}那么称f为G的k-邻点可区别的全染色(简记为k-AVDTC),称min{k|G有k-邻点可区别的全染色}为G的邻点可区别的全色数,记作Xat(G)．本文得到了联图Fn∨Pm的全色数．     

LOCALIZATION THEOREM ON HAMILTONIAN GRAPHS

1IntroductionInthispaper,Weuse[1]forterminologyandnotationnotdefinedhereandconsiderfinitesillWlegraphsonlyThedistancebetweenverticesuandvisdenotedbyd(u,v)-ForeachvertexuEV(G),wedeuotebyN(u)thesetofallverticesofGadjacenttou.ThesubgraphofGinducedbyN(u)U{u}isdenotedbyG(u).IfuveE(G),wedenotebyS(u,v)thenumberofedgesofmaximumstarincludingu5vasaninducedsubgraphinG.Letxai1dybetwoverticesinGwitl1d(x,y)=2,wedefineI(x,y)=IN(x)nN(y)I.LetCbeacycleofGwithafixedcyclicorientation.ForuEV(C),letu be…     

图的笛卡儿积的测地数

对于图G内的任意两点u和v，u-v测地线是指u和v之间的最短路．I（u，v）表示位于u-v测地线上所有点的集合，对于．S∈V（G），I（S）表示所有I（u，v）的并，这里“u,v∈．S．G的测地数g（G）是使I（S）=V（G）的点集．S的最小基数．在这篇文章，我们研究G×K3的测地数和g（G）与g（G×K3）相等的充分必要条件，还给出了T×Km和Cn×Km的测地数，这里T是树．     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

图的Hyper-Wiener指数的综述(英文)

设G是一个图.G的顶点u和v的距离是u和v之间最短路的长度.Wiener指数是G中所有无序顶点对之间距离之和,而Hyper-Wiener指数定义为WW(G)=?∑u,v∈V(G)d(u,v)+?∑u,v∈V(G)d2(u,v),式中的和取遍G的所有顶点对.本文总结了图的Hyper-Wiener指数的最近结论.     

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt(G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K8,n are discussed in this paper. Particularly, the VDIET chromatic number of K8,n are obtained.     

若干图的强染色

图 G(V,E)的一正常 k-染色 σ称为 G(V,E)的 - k-强染色当且仅当对任何两个不同顶点 u和 v,只要d(u,v)≤ 2 ,则 u、v染不同颜色 (这里 d(u,v)表示 u,v之间的距离 ) ,并称 xs(G) =min{ k|存在 G的 - k-强染色 }为 G的强色数 ,本文得到 θ-图 ,Cm,n图 ,Halin图的强色数 xs(G)     

本刊英文版Vol.29(2013),No.8论文摘要(英文)

<正>L(j,k)-number of Direct Product of Path and Cycle Wai Chee SHIU Qiong WU Abstract For positive numbers j and k,an L(j,k)-labeling f of G is an assignment of numbers to vertices of G such that |f(u)-f(v)|≥j if uv∈E(G),and |f(u)-f(v)|≥k if d(u,v)=2.Then the span of f is the difference between the maximum and the minimum numbers assigned by f.The L(j,k)-number of G,denoted byλ_(j,k)(G),is the minimum span over     

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.     

Hamilton连通图的一个充分条件

Let G be a3-connected graph with n vertices.The paper proves that if for each pair of verti-ces u and v of G,d(u,v)=2,has|N(u)∩N(v)|≤α（αis the minimum independent set num-ber),and then max{d(u),d(v)|≥n 1/2,then G is a Hamilton connected graph.