点可迁图的限制边连通度

设S是连通图G的边子集.如果G-S不连通而且不含孤立点,那么称S是G的一个限制边割.G中所有限制边割中最小边数称为G的限制边连通度,记为′(G).限制边连通度是对传统边连通度的推广,而且是计算机互连网络容错性的一个重要度量.点可迁图是一类重要的网络模型.本文证明了如下结论 设G是连通的点可迁图.如果G的点数n4,而且点度k2,那么或者′(G)=2k-2,或者n是偶数,G含三角形且存在整数m2,使得k′(G)=n/m2k-3.     

点可迁图的限制边连通性

3限制边割是连通图的一个边割, 它将此图分离成阶不小于3的连通分支. 图G的最小3限制边割所含的边数称为此图的3限制边连通度, 记作λ\-3(G). 它以图G的3阶连通点导出 子图的余边界的最小基数ξ_3(G)为上界. 如果λ_3(G)=ξ_3(G), 则称图G是极大3限制边连通的 . 已知在某种程度上,3限制边连通度较大的网络有较好的可靠性. 作者在文中证明: 如果k正则连通点可迁图的 围长至少是5, 那么它是是极大3限制边连通的.     

极大3限制边连通二部图的充分条件

设G=(V,E)是一个连通图.称一个边集合S■E是一个k限制边割,如果G-S的每个连通分支至少有k个顶点.称G的所有k限制边割中所含边数最少的边割的基数为G的k限制边连通度,记为λ_k(G).定义ξ_k(G)=min{[X,■]:|X|=k,G[X]连通,■=V(G)\X}.称图G是极大k限制边连通的,如果λ_k(G)=ξ_k(G).本文给出了围长为g>6的极大3限制边连通二部图的充分条件.     

n 阶临界3-边连通图的最大边数

本文中未经说明的术语和记号采自[2].设 G=(V,E)是一个简单图。G 的顶点数记作 n(G),边数记作 m(G),即 n(G)=|V|,m(G)=|E|.假设 G 是3-边连通图.G 的顶点 v(?)V 称为 G 的临界点,如果 G-v 不是3-边连通的;否则称为 G 的非临界点.如果每个 v(?)V 都是 G 临界点,则称 G 是临界3-边连通图.临界3-边连通图类记作 A,A_n 是 A 中所有 n 阶图的集合.假设 G(?)A,则对每个 v∈A,     

超级限制边连通二部图的充分条件

设S是连通图G的一个边割.若G-S不包含孤立点,则称S是G的一个限制边割.图G的最小限制边割的边数称为G的限制边连通度,记为λ'(G).如果图G的限制边连通度等于其最小边度,则称图G是最优限制边连通的,简称λ'-最优的.进一步,如果图G的每个最小限制边割恰好分离出图G的一条边,则称图G是超级限制边连通的,简称超级-λ'的.设G是一个最小度δ(G)≥2的n≥4阶二部图,ξ(G)是G的最小边度.本文证明了(a)若ξ(G)≥(n/2-2)(1+1/δ(G)-1),则G是λ'-最优的;(b)若ξ(G)>(n/2-2)(1+1/δ(G)-1),则G是超级-λ'的,除非图G是K2,n-2,n≥6或是Cartesian积图Kn/4,n/4×K2,其中n≥8且n整除4.最后,论文举例说明该结果是最好可能的.     

周长为4不含3圈的m限制边连通图

m限制边割是连通图的一个边割,它将此图分离成阶不小于m的连通分支刻画了周长为4,不含3圈的m限制边割的图类.     

网络连通性的最优化

简述了极大边连通图和超边连通图;限制边连通度、极大限制边连通图和超限制边连通图的研究进展.     

连通图的谱半径上界

图谱理论是图论研究的重要的领域之一.设图G是n阶简单连通图,具有n顶点和m条边的连通图,p(G)为图G的邻接矩阵的谱半径.利用代数的方法得出两个ρ(G)的上界为:■与■和达到上界的图.     

局部连通图的群Z_3-连通性

本文研究了局部连通图的群连通性的问题.利用不断收缩非平凡Z_3-连通子图的方法,在G是3-边连通且局部连通的无爪无沙漏图的情况下,获得了G不是群Z_3-连通的当且仅当G是K_4或W_5.推广了当G是2-边连通且局部3-边连通时,G是群Z_3-连通的这个结果.     

3限制边连通度与正则因子

设G是一个阶不小于6的k正则连通点可迁图. 如果G不含三角形, 那么图G是极大3限制边连通的, 或者G含有各连通分支都同构于同一个h阶点可迁图的k-1正则因子, 其中2k-2≤h≤3k-5. 唯一的例外是: G是围长等于4 的3正则图.     

模糊图中的割点,割边及块的性质研究

在简单模糊图的基础上引入了模糊子图以及模糊图的割点、割边和块的概念,并讨论了模糊图的割点、割边及其块的一些性质.     

图的3限制性边割

3限制性边割将连通图分离成不连通图，使其各连通分支含有至少3个顶点．含3限制性边割的图在本文中得到刻划．     

正则图的限制性边连通度

将连通图分离成阶至少为二的分支之并的边割称为限制性边割,最小限制性边割的阶称为限制性边连通度. 用λ′(G)表示限制性连通度,则λ′(G)≤ξ(G),其中ξ(G)表示最小边度. 如果上式等号成立,则称G是极大限制性边连通的. 本文证明了当k＞|G|/2时,k正则图G是极大限制性边连通的,其中k≥2, |G|≥4; k的下界在某种程度上是不可改进的.     

On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

关于超欧拉图的一个注记

设G是无向无环的有限图 ,若G有一个生成子图是欧拉图 (Euler) ,则称G是超欧拉图 (Supereulerian) .本文不利用收缩方法 ,直接证明了 :当图G至多差一边有两棵边不相交的生成树时 ,G是超欧拉图或者G有割边 .     

On the 3-restricted edge connectivity of permutation graphs

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G−W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge connectivity of a permutation graph is upper bounded by the so-called minimum k-edge degree. In this paper some sufficient conditions guaranteeing optimal k-restricted edge connectivity and super k-restricted edge connectivity for permutation graphs are presented for k=2,3.     

Directed cut transversal packing for source-sink connected graphs

Concerning the conjecture that in every directed graph, a maximum packing of directed cut transversals is equal in cardinality to a minimum directed cut, a proof is given for the side coboundaries of a graph. This case includes, and is essentially equivalent to, all source-sink connected graphs, for which Schrijver has given a proof. The method used here first reduces the assertion to a packing theorem for bi-transversals. A packing of bi-transversals of the required size is constructed one edge at a time, by maintaining a Hall-like feasibility condition as each edge is added.     

Super Restricted Edge Connectivity of Regular Graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs. Supported by NNSF of China (10271105); Ministry of Science and Technology of Fujian (2003J036); Education Ministry of Fujian (JA03147)