3—连通局部连通无爪图是泛连通图

本文证明了若G是连通、局部连通的无爪图,则G是泛连通图的充要条件为G是3-连通图.这意味着H.J.Broersma和H.J.Veldman猜想成立.     

4连通图的可去边与4连通图的构造

本文引进了４连通图的可去边的概念,,并证明了４连通图Ｇ中不存在可去边的充要条件是Ｇ＝Ｃ５或Ｃ６,同时给出了ｎ阶４连通图的一个新的构造方法．     

一些条件连通图的关系

在这篇文章中，我们主要研究一些条件连通图之间的关系，如上连通，上边连通，超连通和上混合连通．     

有限连通图上随机扩张森林和连通子图的边负相关性

设G为有限连通图．本文研究图G的子图空间G上的三类概率测度,它们分别刻画图的随机扩张树,随机扩张森林和随机连通子图．基于G上均匀扩张树的边负相关性,我们构造G上的一族边负相关的非平凡随机扩张森林和随机连通子图．此外,我们还给出一定条件下图上均匀扩张森林的边负相关性．     

3连通图的可去边数

设 e是 3连通图 G的一条边 ,如果 G- e是某个 3连通图的剖分 ,则称 e是 G的可去边 .本文给出了 3连通图的可去边数依赖于极大半轮的下界以及达到下界的极图 .     

上连通和超连通三次点传递图

图G称为上连通的,若对每个最小割集C,G-C有孤立点,G称为超连通的,若对每个最小割集C,G-C恰有两个连通分支,且其中之一为弧立点,本文刻划了上连通和超连通三次点传递图。     

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.     

临界κ连通图中的点度数

如果在—个κ连通图G中删掉任意一个顶点后得到的图都不再是κ连通,则称G为临界κ连通.Chartrand,Kaugars和Lick证明了每一个临界κ连通图(κ≥2)都含有一个度数小于(3κ-1)/2的顶点.Hamidoune进一步证明了每一个临界k连通图都至少含有两个这样的顶点,并且这一下界是最优的.在本文中,我们证明如果一个临界κ连通图恰好含有两个度数小于(3κ-1)/2的顶点,则这两个顶点的度数一定是κ.     

2-连通无爪图的连通因子

若图G不含有同构于K1,3的导出子图,则称G为一个无爪图.令a和b是两个整数满足2≤a≤b.本文证明了若G是一个含有[a,b]因子的2连通无爪图,则G有一个连通的[a,b 1]因子.     

关于连通图的k阶幂图的几个性质

本文研究了简单连通图的 k 阶幂图的一些性质,给出了有关边连通、局部连通和叶连通的结果,以及有关泛圈和泛连通的结果。     

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.     

A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph.     

图的连通控制数与无赘数的一个不等式

设G是连通图,γ_C(G)和ir(G)分别表示G的连通控制数和无赘数。孙良于1990年证明了γ_c(G)≤4ir(G)—2,同时提出猜想γ_c(G)≤3ir(G)—2。本文进一步研究γ_c(G)与ir(G)的关系,并证得上述猜想成立。     

The Connected Monophonic Number of a Graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x ? y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by m _c (G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance d _m (u, v) from u to v is defined as the length of a longest u ? v monophonic path in G. The monophonic eccentricity e _m (v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad _m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam _m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r <  d, there exists a connected graph G with rad _m G =  r, diam _m G =  d and m _c (G) =  n. Also, if a,b and p are positive integers such that 2 ≤  a <  b ≤  p, then there exists a connected graph G of order p, m(G) =  a and m _c (G) =  b.     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

The Circular Flow Number of a 6-Edge Connected Graph is Less Than Four

We show that every 6-edge connected graph admits a circulation whose range lies in the interval [1,3). Received March 29, 2000 RID="*" ID="*" Supported by NATO-CNR Fellowship; this work was done while the author was visiting the Dept. of Mathematics and Statistics at Simon Fraser University, Canada. RID="†" ID="†" Supported by a National Sciences and Engineering Research Council Research Grant     

Continuous-Time Independent Edge-Markovian Random Graph Process

In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence of the number of isolated nodes and the probability of having no isolated nodes when the initial distribution of the random graph process is stationary distribution, derive the lower limit of the probability in which two arbitrary nodes are connected and the random graph is also connected, and prove that the random graph is almost everywhere connected when the number of nodes is sufficiently large.     

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S _x ? V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S _x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm _x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S _x such that the subgraph induced by S _x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm _x (G). A connected x-detour monophonic set S _x of G is called a minimal connected x-detour monophonic set if no proper subset of S _x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm⁺ _x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm _x (G) = j,dm⁺ _x (G) = k and cdm⁺ _x (G) = l for some vertex x in G.     

双圈连通图的L(2,1)-labelling

给定图G,G的一个L(2,1)-labelling是指一个映射f:V(G)→{0,1,2,…},满足:当dG(u,v)=1时,f(u)-f(v)≥2;当dG(u,v)=2时,f(u)-f(v)≥1.如果G的一个L(2,1)-labelling的像集合中没有元素超过k,则称之为一个k-L(2,1)-labelling.G的L(2,1)-labelling数记作l(G),是指使得G存在k-L(2,1)-labelling的最小整数k.如果G的一个L(2,1)-labelling中的像元素是连续的,则称之为一个no-holeL(2,1)-labelling.本文证明了对每个双圈连通图G,l(G)=△ 1或△ 2.这个工作推广了[1]中的一个结果.此外,我们还给出了双圈连通图的no-hole L(2,1)-labelling的存在性.     

Connected matchings in chordal bipartite graphs

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft (2003) introduced connected matchings and proved that, given a positive integer $k$, determining whether a graph has a connected matching of size at least $k$ is NP-complete. Cameron (2003) proved that this problem remains NP-complete on bipartite graphs, but can be solved in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertex-elimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining–in a totally balanced binary matrix–the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal.