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。     

关于Sumner-Blitch猜想的一个注记

设G是一个图。G的最小度，连通度，控制数，独立控制数和独立数分别用δ，k，γ，i和α表示，图G是3－γ-临界的，如果γ=3，而且G增加任一条边所得的图的控制数为2.Sumner和Blitch猜想：任意连通的3－γ临界图满足i=3，本文证明了如果G是使α＝k 1≤δ的连通3－γ-临界图，那么Sumner-Blitch猜想成立。     

最大临界2-边连通图的结构

假若G是一个2-边连通图,但对G中任一点v,G\{v}不是2-边连通图,则称G为一个临界2-边连通图。具有最大边数的临界2-边连通图称为一个最大临界图。文[1]中,作者给出了p阶临界2-边连通图的边数的最大界f(p),列出了最大临界图结构的不同情况。并且他们猜测已经找出所有这类图。本文将证明他们的猜想是正确的。     

k-临界2k-连通图

图G 称为(n, k)-图, 如果对任一SÍ V(G) (|S|≤k)有k(G-S)=n-|S|, 其中k(G)表示G的连通度. Mader猜想当k≥3时K_2k+2-(1-因子)是惟一的(2k, k)-图. M. Kriesell 解决了k = 3, 4的特殊情形. 对k≥5的一般情形, 证明了该猜想成立.     

K1,4-自由的模κ泛圈图

设G是2-连通的K1，4自由图．本文证明了当δ(G)≥κ 1时，G是模κ泛圈图．这一结果肯定了猜想2，继而也肯定了Thomassen猜想在2-连通图中的正确性．     

关于3正则图的三匹配交猜想(Ⅱ)

Fan和Raspaud 1994年提出如下猜想：任一无桥3正则图必有三个交为空集的完美匹配．本文证明了如下结果：若G是一个圈4-边连通的无桥3正则图，且存在G的一个完美匹配M1使得G—M1恰为4个奇圈的不交并，则存在图G的两个完美匹配M2和M3使得M1∩M2∩M3=Φ。     

分数k-因子临界图的条件

设G是-个连通简单无向图,如果删去G的任意k个项点后的图有分数完美匹配,则称G是分数k-因子临界图．给出了G是分数k-因子临界图的韧度充分条件与度和充分条件,这些条件中的界是可达的,并给出G是分数k-因子临界图的一个关于分数匹配数的充分必要条件．     

直径不超过2的2连通无爪图是哈密顿图

不包含2K_2的图是指不包含一对独立边作为导出子图的图.Kriesell证明了所有4连通的无爪图的线图是哈密顿连通的.本文证明了如果图G不包含2K_2并且不同构与K_2,P_3和双星图,那么线图L(G)是哈密顿图,进一步应用由Ryjá(?)ek引入的闭包的概念,给出了直径不超过2的2连通无爪图是哈密顿图这个定理的新的证明方法.     

Balinski和Russakoff关于分配多面体图连通性猜想的证明

在文[1]中,M.L.Balinski和A.Russakoff猜测,分配多面体图G(P_n)是N(n)-连通的,这里是G(P_n)的正则度数。本文证明了这一猜想,得到G(P_n)的连通度等于N(n)。     

论两种2-棱-连通图

无自圈的极小2-棱-连通图构造已由[1]及[3]给出,最近朱必文又得到了临界2-棱-连通图的构造本文研究了极小2-棱-连通图与临界2-棱-连通图之间的转化关系,从而得到了由前者过渡到后者的一种方法。本文在极小2-棱-连通图构造的基础上首先研究了临界-极小2-棱-连通图的构造,由此得出临界2-棱-连通图的一种非常简洁的递归结     

Contractible Edges in 7-Connected Graphs

An edge of a k-connected graph is said to be a k-contractible edge, if its contraction yields again a k-connected graph. A noncomplete k-connected graph possessing no k-contractible edges is called contraction critical k-connected. Recently, Kriesell proved that every contraction critical 7-connected graph has two distinct vertices of degree 7. And he guessed that there are two vertices of degree 7 at distance one or two. In this paper, we give a proof to his conjecture. The work partially supported by NNSF of China(Grant number: 10171022)     

The New Lower Bound of the Number of Vertices of Degree 5 in Contraction Critical 5-Connected Graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. Let G be a contraction critical 5-connected graph, in this paper we show that G has at least ${\frac{1}{2}|G|}$ vertices of degree 5.     

A new lower bound on the number of trivially noncontractible edges in contraction critical 5-connected graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. An edge of a k-connected graph is called trivially noncontractible if its two end vertices have a common neighbor of degree k. Ando [K. Ando, Trivially noncontractible edges in a contraction critically 5-connected graph, Discrete Math. 293 (2005) 61-72] proved that a contraction critical 5-connected graph on n vertices has at least n/2 trivially noncontractible edges. Li [Xiangjun Li, Some results about the contractible edge and the domination number of graphs, Guilin, Guangxi Normal University, 2006 (in Chinese)] improved the lower bound to n+1. In this paper, the bound is improved to the statement that any contraction critical 5-connected graph on n vertices has at least trivially noncontractible edges.     

Vertices of Degree 5 in a Contraction Critically 5-connected Graph

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Local structure of 5- and 6-connected graphs

We prove that if graph on n vertices is minimally and contraction critically 5-connected, then it has 4n/7 vertices of degree 5. We also prove that if graph on n vertices is minimally and contraction critically 6-connected, then it has n/2 vertices of degree 6. Bibliography: 7 titles.     

A constructive characterization of contraction critical 8-connected graphs with minimum degree 9

An edge of a $k$-connected graph is said to be $k$-contractible if its contraction results in a $k$-connected graph. A $k$-connected graph without $k$-contractible edge is said to be contraction critically $k$-connected. Y. Egawa and W. Mader, independently, showed that the minimum degree of a contraction critical $k$-connected graph is at most $\frac{5k}{4}?1$. Hence, the minimum degree of a contraction critical 8-connected graph is either 8 or 9. This paper shows that a graph $G$ is a contraction critical 8-connected graph with minimum degree 9 if and only if $G$ is the strong product of a contraction critical 4-connected graph $H$ and $K_2$.     

Some structural properties of minimally contraction-critically 5-connected graphs

An edge of a k-connected graph is said to be k-removable (resp. k-contractible) if the removal (resp. the contraction ) of the edge results in a k-connected graph. A k-connected graph with neither k-removable edge nor k-contractible edge is said to be minimally contraction-critically k-connected. We show that around an edge whose both end vertices have degree greater than 5 of a minimally contraction-critically 5-connected graph, there exists one of two specified configurations. Using this fact, we prove that each minimally contraction-critically 5-connected graph on n vertices has at least vertices of degree 5.