两个不交图的联图的最小圈基长度

这篇文章中,我们分两种情形分别给出了计算两个不交图的联图的最小圈基长度的公式.作为它们的应用,我们给出了计算n个相同的图的联图以及完全r-部图等图的最小圈基长度的公式.     

循环图圈空间的最小基

For most of circular graph the length of the minimum cycle basis is given.For the others a bound of the length of the minimum cycle basis is given and the given bound is reached.     

图的最小完整度

主要研究了图的完整度,并给出若干关于完整度的结果. 对于所有的顶点数和边数都给定的连通图类,如何确定该图类中完整度最小的图. 同时研究了对于顶点数和完整度都给定的连通图类,如何确定该图类中边数最多的图的问题. 这些结果为图的最小完整度的优化设计提供了理论和方法.     

局部大边宽嵌入图的最小圈基

利用Thomassen等人在大边宽嵌入方面的工作,给出局部大边宽嵌入的定义,并运用线性代数和相异代表系的知识,证明了局部大边宽嵌入图的最小圈基.     

点染色3-边划分图的最小度

Addrio-Berry L[1]已经证明了最小度至少为1000的图可以点染色3-边划分,在本文中,我们将其结果改进到了最小度至少为662.     

原始图与三次图的可圈度

本文综述关于原始图与三次图的可圈度的近期结果并提出一些未解决的问题。     

图中相互独立的4圈和含4个点的路

设k是一个正整数，G是一个顶点数为|G|=4k的图. 假设σ\-2(G)≥4k-1, 则G有一个支撑子图含k-1个4圈和一条顶点数为4的路，使得所有这些圈和路都是相互独立的. 设G=(V\-1,V \-2;E)是一个二分图使得|V\-1|=|V\-2|=2k. 如果对G中每一对满足x∈V\-1和y∈V\-2的不 相邻的顶点x和y 都有d(x)+d(y)≥2k+1, 则G包含k-1个相互独立的4圈和一条顶点数为4的路，使得所有这些圈和路都是相互独立的，并且此度条件是最好的.     

k圈图的最大Laplace分离度

设$ G $是一个$ n $阶$ k $圈图, $ k $圈图为边数等于顶点数加$ k-1 $的简单连通图。$ \mu_{1}(G) $、$ \mu_{2}(G) $分别记为图$ G $的Laplace矩阵的最大特征值和次大特征值, 图$ G $的Laplace分离度定义为$ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $。本文研究了给定阶数的$ k $圈图的最大Laplace分离度, 并刻画了相应的极图, 其结果推广了已有当$ k=1, 2, 3 $时的结论。     

k圈图的最大Laplace分离度

设$ G $是一个$ n $阶$ k $圈图, $ k $圈图为边数等于顶点数加$ k-1 $的简单连通图。$ \mu_{1}(G) $、$ \mu_{2}(G) $分别记为图$ G $的Laplace矩阵的最大特征值和次大特征值, 图$ G $的Laplace分离度定义为$ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $。本文研究了给定阶数的$ k $圈图的最大Laplace分离度, 并刻画了相应的极图, 其结果推广了已有当$ k=1, 2, 3 $时的结论。     

最小封堵圈的扩张算法设计

在发生突发事件之后,及时制定有效的封堵方案才能掌握抓捕嫌疑犯的主动权.提出了对逃跑罪犯实施完全封堵的最小包围圈的计算机算法,通过规则图形的仿真演示,验证了该算法的正确性.最后以某市案发时候的警力分布和交通图作为实例,模拟了对嫌疑犯实施封堵时计算最小包围圈的最优方案.结果证明该算法具有简洁、可靠的优点.     

Vertex-Disjoint Triangles in Claw-Free Graphs with Minimum Degree at Least Three

. Our main result is as follows: For any integer , if G is a claw-free graph of order at least and with minimum degree at least 3, then G contains k vertex-disjoint triangles unless G is of order and G belongs to a known class of graphs. We also construct a claw-free graph with minimum degree 3 on n vertices for each such that it does not contain k vertex-disjoint triangles. We put forward a conjecture on vertex-disjoint triangles in -free graphs. Received: November 21, 1996/Revised: Revised February 19, 1998     

最小次数至少为4的超欧拉图

设Ｇ是２－边－连通的ｎ阶图。假设对任何的的最小边割集Ｅ等于包含于Ｅ（Ｇ）且│Ｅ│≤３,Ｇ－Ｅ的每个分支的阶至少为ｎ／５,则或者Ｇ是一个超欧拉图或者Ｇ有５个互不相交的阶数为ｎ／５连通分支,当这５个分支都收缩时,Ｇ收缩为Ｋ２,３,这个结果推广了蔡小涛,Ｐ．Ａ．Ｃａｔｌｉｎ,Ｆ．Ｊａｅｇｅｒ和Ｈ．Ｊ．Ｌａｉ等人关于超欧拉图的结果。     

Restrained Domination in Claw-Free Graphs with Minimum Degree at Least Two

Let G = (V, E) be a graph. A set \({S\subseteq V}\) is a restrained dominating set if every vertex in V ? S is adjacent to a vertex in S and to a vertex in V ? S. The restrained domination number of G, denoted γ _r (G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is claw-free with minimum degree at least two and \({G\notin \{C_{4},C_{5},C_{7},C_{8},C_{11},C_{14},C_{17}\}}\) , then \({\gamma_{r}(G)\leq \frac{2n}{5}.}\)     

大次和的1－坚韧图中的最长圈

给一个图Ｇ，定义，是Ｇ的无关集，是Ｇ中使的无关集，本文证明了：设Ｇ是ｎ阶１－坚韧图，如果σｓ３≥ｎ。，则Ｇ包含长度至少为ｍｉｎ的圈。这个结果推广了若干已知结果，也解决了Ｂｒｏｅｒｓｍａ－Ｈｅｕｖｅｌ－Ｖｅｌｄｍａｎ所提猜想的一个特例．     

Decomposing Graphs of High Minimum Degree into 4‐Cycles

If a graph G decomposes into edge‐disjoint 4‐cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least , where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least . On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree , but having no decomposition into edge‐disjoint 4‐cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4‐cycles are sufficient when G has minimum degree at least .     

Maximizing H‐Colorings of Connected Graphs with Fixed Minimum Degree

For graphs G and H , an H‐coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H , which connected n‐vertex graph with minimum degree δ maximizes the number of H‐colorings? We show that for nonregular H and sufficiently large n , the complete bipartite graph is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph is the unique k‐connected graph that maximizes the number of H‐colorings among all k‐connected graphs. Finally, we show that this conclusion does not hold for all regular H by exhibiting a connected n‐vertex graph with minimum degree δ that has more ‐colorings (for sufficiently large q and n ) than .     

Characterizing Graphs with Crossing Number at Least 2

Our main result includes the following, slightly surprising, fact: a 4‐connected nonplanar graph G has crossing number at least 2 if and only if, for every pair of edges having no common incident vertex, there are vertex‐disjoint cycles in G with one containing e and the other containing f.