Highly edge-connected regular graphs without large factorizable subgraphs

We construct highly edge-connected $r$-regular graphs of even order which do not contain $r?2$ pairwise disjoint perfect matchings. When $r$ is a multiple of 4, the result solves a problem of Thomassen [4].     

最大边连通和super-边连通超图的充分条件

设H是连通超图。若超图H的边连通度等于其最小度,则称H是最大边连通的。若超图H的每个最小边割总是由关联于某个最小度顶点的边集所构成,则称H是super-边连通的。首先给出一致线性超图是最大边连通超图的度序列条件。其次,给出一致线性超图是super-边连通超图的度条件。这些结果分别推广了Dankelmann和Volkmann（1997）以及Hellwig和Volkmann（2005）在图上的相关结论。     

关于分数因子两个开问题的解答

本文指出极小连通二部分数1-因子不一定是极小2-连通图.研究了σ₂（G）与分数k-因子存在性之间的关系,指出存在一个特例在满足阶数n≥4k-5,δ（G）≥k且σ₂（G）≥n条件下,图G不存在分数k-因子.     

Maximally edge-connected and vertex-connected graphs and digraphs: A survey

Let D be a graph or a digraph. If δ(D) is the minimum degree, λ(D) the edge-connectivity and κ(D) the vertex-connectivity, then κ(D)?λ(D)?δ(D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if λ(D)=δ(D) and maximally vertex-connected if κ(D)=δ(D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively.     

Configurations containing a given linear hypergraph

We investigate the minimum order of a linear r‐regular k‐uniform hypergraph, also known as an ‐combinatorial configuration, which contains a given linear k‐uniform hypergraph of maximum (vertex) degree at most r.     

Asymptotic normality in random graphs with given vertex degrees

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and Röllin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.     

A [k,k+1]-factor containing given Hamiltonian cycle

Letk⩾2 be an integer and let G be a graph of ordern with minimum degree at leastk, n⩾8k -16 for evenn and n⩾6k - 13 for oddn. If the degree sum of each pair of nonadjacent vertices of G is at least n, then for any given Hamiltonian cycleC. G has a [k, k + 1]-factor containingC Preject supported partially by an exchange program between the Chinese Academy of Sciences and the Japan Society for Promotion of Sciences and by the National Natural Science Foundation of China (Grant No. 19136012)     

Hamiltonian graphs of given order and minimum algebraic connectivity

In this paper, we study the algebraic connectivity of a Hamiltonian graph, and determine all Hamiltonian graphs whose algebraic connectivity attain the minimum among all Hamiltonian graphs on n vertices.     

二分图中含有完美对集的2因子

该文证明若G是2n阶均衡二分图，δ(G)≥(2n-1)/3，则对任何正整数k，n≥4k时，任给G的一个完美对集M，G中存在一个包含M的所有边的恰含k个分支的2 因子(k=1，n=5且δ(G)=3除外）. 特别k=2时，在条件n≥5且δ(G)≥(n+2)/2下，结论也成立. 这里所给的δ(G)的下界是最好的可能.      

On connected factors inK 1,3-free graphs

A graph is said to beK _1,3-free if it contains noK _1,3 as an induced subgraph. It is shown in this paper that every 2-connectedK _1,3-free graph contains a connected [2,3]-factor. We also obtain that every connectedK _1,3-free graph has a spanning tree with maximum degree at most 3. This research is partially supported by the National Natural Sciences Foundation of China and by the Natural Sciences Foundation of Shandong Province of China.     

ON CONNECTED FACTORS IN K_(1,3)-FREE GRAPHS

1.IntroductionWeconsiderfinitesimplegraphs,andfollowBondyandMurtyl'lforgeneralterminologyandnotation.LetG=(V(G),E(G))beagraphwithavertexsetV(G)andanedgesetE(G).ForavertexvEV(G),N(v,G)denotesthesetofneighborsofvinG,anddG(v)=IN(v,G)l.Foravertexsubset(resp.subgraph)HofG,G--HdenotesthesubgraphofGobtainedfromGbydeletingthevenicesinHtogetherwiththeedgesincidelltwiththem.IfAisanedgesubsetofGandHasubgraphofG,thenH AdenotesthesubgraphofGobtainedfromHbyaddingtheedgesinAtogetherwiththeven…     

On the order of certain close to regular graphs without a matching of given size

A graph G is a {d, d+k}-graph, if one vertex has degree d+k and the remaining vertices of G have degree d. In the special case of k = 0, the graph G is d-regular. Let k, p ⩾ 0 and d, n ⩾ 1 be integers such that n and p are of the same parity. If G is a connected {d, d+k{-graph of order n without a matching M of size 2|M| = n − p, then we show in this paper the following: If d = 2, then k ⩾ 2(p + 2) and

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

Orientations of graphs avoiding given lists on out-degrees

Let $G$ be a graph and $F:V\left(G\right)\to 2^{\mathbb{N}}$ be a mapping. The graph $G$ is said to be $F$- avoiding if there exists an orientation $O$ of $G$ such that $d_O^{+}\left(v\right)\notin F\left(v\right)$ for every $v\in V\left(G\right)$, where $d_O^{+}\left(v\right)$ denotes the out-degree of $v$ in the directed graph $G$ with respect to $O$. In this paper it is shown that if $G$ is bipartite and $\mid F\left(v\right)\mid \le d_G\left(v\right)/2$ for every $v\in V\left(G\right)$, then $G$ is $F$-avoiding. The bound $\mid F\left(v\right)\mid \le d_G\left(v\right)/2$ is best possible. For every graph $G$, we conjecture that if $\mid F\left(v\right)\mid \le \frac{1}{2}\left(d_G\left(v\right)-1\right)$ for every $v\in V\left(G\right)$, then $G$ is $F$-avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

Minimizing algebraic connectivity over graphs made with some given blocks

This article describes the structure of the graph minimizing the algebraic connectivity among all connected graphs made with some given blocks with fixed number of pendant blocks, the blocks that have exactly one point of articulation. As an application, we conclude that over all graphs made with given blocks, the algebraic connectivity is minimum for a graph whose block structure is a path.     

Cycles and paths in edge‐colored graphs with given degrees

Sufficient degree conditions for the existence of properly edge‐colored cycles and paths in edge‐colored graphs, multigraphs and random graphs are investigated. In particular, we prove that an edge‐colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to ?(n+1)/2? has properly edge‐colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge‐colored paths and hamiltonian paths between given vertices are considered as well. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 63–86, 2010     

On triconnected and cubic plane graphs on given point sets

Let S be a set of n4 points in general position in the plane, and let h<n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max{3n/2,n+h−1} edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n4 is even and hn/2+1. The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.