不含5-圈的平面图的无圈边着色

图G的一个无圈边着色是一个正常的边着色且不含双色的圈.图G的无圈边色数是图G的无圈边着色中所用色数的最小者.本文用反证法得到了不含5-圈的平面图G的无圈边色数的一个上界.     

具有两个同阶轨道的双轨道图的圈边连通度

一个边割被称为圈边割,如果该边割能分离图的两个不同圈.如果一个图有圈边割,称该图为圈边可分离的.一个圈边可分离图G的最小圈边割的阶数被称为圈边连通度,记作cλ（G）.定义：ζ（G）=min{w（X）｜X导出G的最短圈},其中w（X）为端点分别在X和V（G）-X中的边的数目.如果一个圈边可分离图G使得cλ（G）=ζ（G）成立,称该图是圈边最优的.Tian和Meng在文章[11]以及Yang et al在文章[15]中研究了两种不同的双轨道图的圈边最优性.本文我们将研究具有两个同阶轨道的双轨道图的圈边连通度.     

不含4-,6-圈和相交三角形的平面图的无圈边色数

图G的一个边染色φ:E(G)→{1,2,…,k},若满足任意相邻边都染不同的颜色,且图G不存在双色圈,则称φ为图G的一个无圈k-边染色.图G的无圈边色数χ’_α(G)为使得图G有一个无圈k-边染色的最小正整数k.本文主要证明了对于无4-,6-圈且3-圈与3-圈不相交的平面图G,若Δ(G)≥9,则χ’_α(G)≤Δ(G)+1.     

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.     

模糊图的g-割点和g-割边的性质

模糊图理论是一门建立在模糊集理论与经典图论基础上的模糊数学分支,其目的是为系统工程、网络设计、计算机科学等领域中的不确定性信息提供分析模型。本文首先引入了模糊图的g-割点和g-割边的概念,其次研究了模糊树、完全模糊图、模糊圈的g-割点和g-割边的相关性质,最后讨论了其在通信网络方面的应用。本文的研究为寻找通信网络中关键设备及线路提供理论依据,有利于更精确地检测和维护通信系统的稳定性。     

图的围长与无圈边色数之间的关系

对于一个图G的正常边着色，如果此种边着色使得该图没有2—色的圈，那么这种边着色被称为是G的无圈边着色．用d(G)表示图G的无圈边色数，即G的无圈边着色中所使用的最小颜色数．Alon N，Sadakov B and Zaks A在[1]中有如下结果：对于围长至少是2000△(G)log△(G)的图G，有d(G)≤△ 2，其中△是图G的最大度．我们改进了这个结果，得到了如下结论：对于围长至少是700△(G)log△(G)的图G，有d(G)≤△ 2．     

λ_k最优图的充分条件

介绍λk最优图的概念,通过考察图中顶点的邻域和k阶连通子图之间的关系,给出了图是λk最优的一些充分条件.     

图的割宽问题

本文建立了Ｈａｒｐｅｒ型割宽下界估计式，由此求出了轮形图Ｗｎ、完全二部图Ｋ（ｍ，ｎ）、圈幂Ｃｎｒ、格子图：Ｐｍ×Ｐｎ、Ｐｍ×Ｃｎ、Ｃｍ×Ｃｎ以及乘积图：Ｋｍ×Ｐｎ、Ｋｍ×Ｃｎ、Ｃｍｓ×Ｃｎｒ、Ｋｍ×Ｋｎ和强乘积图Ｐｍ　Ｐｎ的割宽。     

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.     

图的邻点可区别无圈边染色的一个界

图G的一个正常边染色被称作邻点可区别无圈边染色,如果G中无二色圈,且相邻点关联边的色集合不同.应用概率的方法得到了图G的一个邻点可区别无圈边色数的上界,其中图G为无孤立边的图.     

The existence of even regular factors of regular graphs on the number of cut edges

For any even integer k and any integer i, we prove that a （kr ＋i）-regular multigraph contains a k-factor if it contains no more than kr - 3k/2＋ i ＋ 2 cut edges, and this result is the best possible to guarantee the existence of k-factor in terms of the number of cut edges. We further give a characterization for k-factor free regular graphs.     

On graphs with cut vertices and cut edges

Let G（n,k,t） be a set of graphs with n vertices,k cut edges and t cut vertices.In this paper,we classify these graphs in G（n,k,t） according to cut vertices,and characterize the extremal graphs with the largest spectral radius in G（n,k,t）.     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.     

A property of weighted graphs without induced cycles of nonpositive weights

It has been conjectured [B. Xu, On signed cycle domination in graphs, Discrete Math. 309 (4) (2009) 1007–1012] that if there is a mapping from the edge set of a 2-connected graph G to {−1,1} such that for each induced subgraph, that is a cycle, the sum of all numbers assigned to its edges by this mapping is positive, then the number of all those edges of G to which 1 is assigned, is more than the number of all other edges of G. This conjecture follows from the main result of this note: If a mapping assigns integers as weights to the edges of a 2-connected graphGsuch that for each edge, its weight is not more than 1 and for each cycle which is an induced subgraph ofG, the sum of all weights of its edges is positive, then the sum of all weights of the edges ofGalso is positive. A simple corollary of this result is the following: If?is a mapping from the edge set of a 2-connected graphGto a set of real numbers such that for each cycleCofG, ∑_e∈E(C)?(e)>0, then∑_e∈E(G)?(e)also is positive.     

Periods in missing lengths of rainbow cycles

A cycle in an edge‐colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge‐colored graph G, define Then ??(G) is a monoid with respect to the operation n°m=n+ m?2, and thus there is a least positive integer π(G), the period of ??(G), such that ??(G) contains the arithmetic progression {N+ kπ(G)|k?0} for some sufficiently large N. Given that n∈??(G), what can be said about π(G)? Alexeev showed that π(G)=1 when n?3 is odd, and conjectured that π(G) always divides 4. We prove Alexeev's conjecture: Let p(n)=1 when n is odd, p(n)=2 when n is divisible by four, and p(n)=4 otherwise. If 2<n∈??(G) then π(G) is a divisor of p(n). Moreover, ??(G) contains the arithmetic progression {N+ kp(n)|k?0} for some N=O(n²). The key observations are: If 2<n=2k∈??(G) then 3n?8∈??(G). If 16≠n=4k∈??(G) then 3n?10∈??(G). The main result cannot be improved since for every k>0 there are G, H such that 4k∈??(G), π(G)=2, and 4k+ 2∈??(H), π(H)=4. © 2009 Wiley Periodicals, Inc. J Graph Theory     

Balloons,cut‐edges,matchings, and total domination in regular graphs of odd degree

A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}$ be the family of connected (2r+1)‐regular graphs with n vertices, and let ${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$. For ${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$, we prove the sharp inequalities c(G)?[r(n?2)?2]/(2r²+2r?1)?1 and α′(G)?n/2?rb/(2r+1). Using b?[(2r?1)n+2]/(4r²+4r?2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ_t(G) of a cubic graph, we prove γ_t(G)?n/2?b(G)/2 (except that γ_t(G) may be n/2?1 when b(G)=3 and the balloons cover all but one vertex). With α′(G)?n/2?b(G)/3 for cubic graphs, this improves the known inequality γ_t(G)?α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010     

Eulerian partial duals of plane graphs

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half‐edge orientations of medial graphs.     

点可迁图中的正则因子

设图G是一个K-正则连通点可迁图.如果G不是极大限制性边连通的,那么G含有一个(k-1)-因子,它的所有分支都同构于同一个阶价于k和2k-3之间的点可迁图.此结果在某种程度上加强了Watkins的相应命题:如果k正则点可迁图G不是k连通的,那么G有一个因子,它的每一个分支都同构于同一个点可迁图.