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.     

Characterization of graphs with infinite cyclic edge connectivity

In this paper, we characterize the graphs with infinite cyclic edge connectivity. Then we design an efficient algorithm to determine whether a graph has finite cyclic edge connectivity or infinite cyclic edge connectivity.     

无向Kautz图的超级限制边连通性

限制边连通度作为边连通度的推广,是计算机互连网络可靠性的一个重要度量.Superλ-′是比限制边连通度更精确的一个网络可靠性指标.一个图是Superλ-′的,如果它的任一最小限制边割都孤立一条有最小边度的边.本文考虑一类重要的网络模型-无向K autz图UK(d,n)的限制边连通度λ,′证明了当d 3,n 2时,λ(′UK(d,n))=4d-4,并进一步指出此时的UK(d,n)是Superλ-′的.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

模糊连通度与模糊核度

本文给出了衡量模糊连通性的三个工量：模糊连通度，模糊边连通度与模糊核度及其相关的性质。与普通图连通性的分析相比较，由于考虑了模糊性，这三个量能更好，更深入地刻划出不同的图在连通性方面的微妙差异。     

Minimally (k,k)‐edge‐connected graphs

For an integer l > 1, the l‐edge‐connectivity of a connected graph with at least l vertices is the smallest number of edges whose removal results in a graph with l components. A connected graph G is (k, l)‐edge‐connected if the l‐edge‐connectivity of G is at least k. In this paper, we present a structural characterization of minimally (k, k)‐edge‐connected graphs. As a result, former characterizations of minimally (2, 2)‐edge‐connected graphs in [J of Graph Theory 3 (1979), 15–22] are extended. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 116–131, 2003     

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.     

Augmenting the Edge‐Connectivity of a Hypergraph by Adding a Multipartite Graph

Given a hypergraph, a partition of its vertex set, and a nonnegative integer k, find a minimum number of graph edges to be added between different members of the partition in order to make the hypergraph k‐edge‐connected. This problem is a common generalization of the following two problems: edge‐connectivity augmentation of graphs with partition constraints (J. Bang‐Jensen, H. Gabow, T. Jordán, Z. Szigeti, SIAM J Discrete Math 12(2) (1999), 160–207) and edge‐connectivity augmentation of hypergraphs by adding graph edges (J. Bang‐Jensen, B. Jackson, Math Program 84(3) (1999), 467–481). We give a min–max theorem for this problem, which implies the corresponding results on the above‐mentioned problems, and our proof yields a polynomial algorithm to find the desired set of edges.     

On the existence of edge cuts leaving several large components

We characterize graphs of large enough order or large enough minimum degree which contain edge cuts whose deletion results in a graph with a specified number of large components. This generalizes and extends recent results due to Ou [Jianping Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005), 365-371] and Zhang and Yuan [Z. Zhang, J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005), 128-134].     

围长为3的点可迁图的3限制边连通度

设G是阶至少为6的k正则连通图.如果G的围长等于3,那么它的3限制边连通度 λ3(G)≤3k-6.当G是3或者4正则连通点可迁图时等号成立,除非G是4正则图并且 λ3(G)=4.进一步,λ3(G)=4的充分必要条件是图G含有子图K4.     

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.     

P_n~2和P_n~(n-1)的均匀邻强边色数

对阶至少为3的简单连通图G的k-正常边染色法f,若对任意uv∈E(G)有C(u)≠C(v),Ei-Ej 1,i,j=1,2,…,k.其中C(u)={f(uv)uv∈E(G)},Ei={uv f(uv)=i,uv∈E(G)},则称f为G的一k-均匀邻强边染色,简称k-EASEC.并称χe′as(G)=min{k k-EASEC of G}为G的均匀邻强边色数.给出了图Pn2与Pnn-1的均匀邻强边色数.     

Extra edge connectivity and isoperimetric edge connectivity

An edge set S of a connected graph G is a k-extra edge cut, if G-S is no longer connected, and each component of G-S has at least k vertices. The cardinality of a minimum k-extra edge cut, denoted by λ_k(G), is the k-extra edge connectivity of G. The kth isoperimetric edge connectivity γ_k(G) is defined as , where ω(U) is the number of edges with one end in U and the other end in . Write β_k(G)=min{ω(U):U⊂V(G),|U|=k}. A graph G with is said to be γ_k-optimal.In this paper, we first prove that λ_k(G)=γ_k(G) if G is a regular graph with girth g?k/2. Then, we show that except for K_3,3 and K₄, a 3-regular vertex/edge transitive graph is γ_k-optimal if and only if its girth is at least k+2. Finally, we prove that a connected d-regular edge-transitive graph with d?6e_k(G)/k is γ_k-optimal, where e_k(G) is the maximum number of edges in a subgraph of G with order k.     

P_m×P_n的邻强均匀边色数

图G的一个k-正常着色满足相邻的点所关联的边的色集合不同,且任两色的边数之差不超过1称为G的k-邻强均匀边染色,图G邻强均匀边染色中最小的k称为图G的邻强均匀边色数.本文得到了P_m×P_n的邻强均匀边色数.     

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 Group Connectivity of Graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z ₃-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z ₃-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z ₃-flow. Devos proposed a stronger version problem by asking if every such graph is Z ₃-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z ₃-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z ₃-connected. It is a partial result to Jaeger’s Z ₃-connectivity conjecture. Received: May 23, 2006. Final version received: January 13, 2008     

周长为3的m限制边割连通图

如果连通图的G存在边割S,使得G-S的每一个连通分支都含有至少m个顶点,则称图G是m限制边连通的.本文刻画了周长为3的m限制边连通图.     

给定边连通度的图的最小距离谱半径(英文)

本文研究了边连通度为r的n阶连通图中距离谱半径最小的极图问题,利用组合的方法,确定了K(n-1,r)为唯一的极图,其中K(n-1,r)是由完全图K_(n-1)添加一个顶点v以及连接v与K_(n-1)中r个顶点的边所构成.上述结论推广了极图理论中的相关结果.     

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)     

Some Remarks on Rainbow Connectivity

An edge (vertex) colored graph is rainbow‐connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colors. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colors needed for a rainbow edge (vertex) coloring of G. In this article, we propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several known results, as well as some new ones.