完全偶图的圈基内插性质

图G的一个圈基的长度是该圈基的所有圈的长度之和。设C^-、C^ 分别是G的最小、最大圈基长度,如果对任一偶数C,C^-＜C＜C^ ,都存在G的一个长为C的圈基,则称G具有偶圈基内插性质,本证明了,完全偶图Km,n具有偶圈基内插性质。     

汉米尔顿图泛圈性的奥尔型条件

本文所说的图是简单图,未定义的术语见[1,2].n 阶图 G,n≥3,若有长为 n 的圈,则说 G 是汉米尔顿图;若对每个 k,3≤k≤n,G 含有长为 k 的圈,则说 G 是泛圈图.定理1.在 n 阶图 G 中,若对任何点对 x,y∈V(G),xy(?)E(G),都有 d(x)+d(y)≥n,则 G 是汉米尔顿图.     

一类几乎唯一泛圈图

设G是阶为n的简单Hamilton图．若存在m(3(?)m相似文献   

关于几乎唯一泛圈图

设G是阶为n的简单Hamilton图．若存在m(3(?)m＜n)使对每个l∈{3,4,…,n} -{m},G恰有一个长为l的圈且不含长为m的圈,则称G是几乎唯一泛圈图,用(?)k表示具有n k条边和恰有1/2(k 1)(k 2)个圈的简单H图的集合,用(?)_k~*表示具有n k条边恰有2~k k个圈的简单外可平面H图的集合,本文确定了(?)_k和(?)_k~*中所有几乎唯一泛圈图,并证明这些图都是简单MCD图,本文还构造了50个含有同胚于K_4的子图的几乎唯一泛圈图,并提出了若干问题和猜想。     

一类新的泛圈图

本文所说的图都是简单无向图。未定义的术语和记号参见[2]。设 G=(V,E)的 n 阶图(n≥3),若 G 中含有 Hamilton 圈,则称 G 是 H-图。若G 中含有从3到 n 的所有长度的圈,则称 G 为泛圈图。如下两个定理是众所周知的。定理1 (Ore,1960)。若在 n 阶图 G 中,有uv(?)E(G)(?)d(u) d(v)≥n,则 G 是 H-图。     

均衡二部图中点不交的4-圈和6-圈（英文）

设G=(V_1,V_2,E)是一个均衡二部图满足|V_1|=|V_2|=n.令δ_(1,1)(G)=min{d(x)+d(y)|x∈V_1,Y∈V_2}.Amar猜想对任意的s个整数(n_1,n_2,…,n_s),n=n_1+n_2+…+n_s,其中n_i≥2.若δ_(1,1)(G)≥n+s,则G含s个点不交的圈,其长分别为2n_1,2n_2,…,2n_s(见[Discrete Math.,1986,58(1):1-10]).本文证明了若一个点数为4k的均衡二部图G满足δ_(1,1)(G)≥2k+4(k≥3),则G含k-3个4-圈和2个6-圈使得所有这些圈都是点不交的.     

Hamiltonian[k,k+1]-因子

本文考虑n/2-临界图中Hamiltonian[k,k+1]-因子的存在性。Hamiltonian[k,k+1]-因子是指包含Hamiltonian圈的[k,k+1]-因子;给定阶数为n的简单图G,若δ(G)≥n/2而δ(G\e)相似文献   

一类完全三部图的色唯一性

用P(G,λ)表示简单图G的色多项式.设G是一个给定的简单图,若对任意简单图H,当P(H,λ)=P(G,λ)时都有H和G同构(记为H≌G),则称图G是色唯一的.本文证明了以下结果:设n,k,△都为非负整数,其中k≥0,△∈{4,5},若n≥1/3k~2+1/3△~2-1/3k△-1/3k-1/3△+4/3,则完全三部图K(n,n+△,n+k)是色唯一的.同时还给出了一个猜想.     

2—连通正则二部图的长圈

Dirac 定理指出:若 G 是 n 个顶点的2-连通图,(?){d(x)}≥k,则 G 有长至少为 min(2k,n)的圈(见[1]).‖本文把 Dirac 定理应用到2-连通正则二部图,得到如下的结果:定理1 设 G 是2-连通 k-正则二部图,G 的顶点数为 n,则 G 有长至少为 min(4k,n)的圈(k≥2).‖     

谱半径前六位的n阶单圈图

恰含一个圈的简单连通图称为单圈图。Cn记n个顶点的圈。△（i,j,κ）记C3的三个顶点上分别接出i,j,κ条悬挂边所得的图，其中i≥j≥κ≥0.Sl^n-l记Cl的某一顶点上接出n-l条悬挂边所得到的图。△（n-4 1,0,0)记△（n-4,0,0)的某个悬挂点上接出一条悬挂边所得到的图。本文证明了：若把所有n(n≥12）阶单圈图按其最大特征值从大到小的顺序排列，则排在前六位的依次是S3^n-3，△（n-4,1,0),△（n-4 1,0,0),S4^n-4,△（n-5,2,0),△（n-5,1,1)。     

On the approximability of the minimum strictly fundamental cycle basis problem

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P=NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log²nloglogn) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.     

The Basis Number of Some Special Non-Planar Graphs

The basis number of a graph G was defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. He proved that for m, n 5, the basis number b(K _m,n ) of the complete bipartite graph K _m,n is equal to 4 except for K _6,10, K _5,n and K _6,n with n = 5, 6, 7, 8. We determine the basis number of some particular non-planar graphs such as K _5,n and K _6,n , n = 5, 6, 7, 8, and r-cages for r = 5, 6, 7, 8, and the Robertson graph.     

The basis number of the composition of theta graphs with stars and wheels

he basis number, b(G), of a graph G is defined to be the least integer k such that G has a k-fold basis for its cycle space. In this paper we investigate the basis number of the composition of theta graphs with stars and wheels. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Minimum cycle bases of Halin graphs

Halin graphs are planar 3‐connected graphs that consist of a tree and a cycle connecting the end vertices of the tree. It is shown that all Halin graphs that are not “necklaces” have a unique minimum cycle basis. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 150–155, 2003     

The basis number of the Cartesian product of stars and wheels with different ladders

The basis number of a graph G is defined to be the least positive integer d such that G has a d-fold basis for the cycle space of G. We investigate the basis number of the cartesian product of stars and wheels with ladders, circular ladders and Möbius ladders.     

Classes of cycle bases

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle bases.In this paper, we offer an unitary classification accommodating these three classes and further including the following four relevant classes: 2-bases (or planar bases), weakly fundamental cycle bases, totally unimodular cycle bases, and integral cycle bases. The classification is complete in that, for each ordered pair (A,B) of classes considered, we either prove that A⊆B holds for every graph or provide a counterexample graph for which A?B. The seven notions of cycle bases are distinct (either A?B or B?A is exhibited for each pair (A,B)).All counterexamples proposed have been designed to be ultimately effective in separating the various algorithmic variants of the MCB problem naturally associated to each one of these seven classes. Furthermore, we provide a linear time algorithm for computing a minimum 2-basis of a graph. Finally, notice that the resolution of the complexity status of some of the remaining three classes would have an immediate impact on practical applications, as for instance in periodic railway timetabling, only integral cycle bases are of direct use.     

联图的圈基

MacLane于1937年给出了圈基方面的重要定理: 图G是平面图, 当且仅当图G有2-重基. 连通图G_1和G_2的联图G_1\vee G_2指的是在它们的不交并G_1\bigcup G_2上添加边集(u,v)|u\in V(G_1), v\in V(G_2). 对G_1和G_2的联图G_1\vee G_2的圈基重数进行了研究, 得到了一个上界, 改进了Zare的结果. 并在此基础之上, 进一步得到特殊联图C_m\vee C_n的圈基重数的一个上界.     

Edge-swapping algorithms for the minimum fundamental cycle basis problem

We consider the problem of finding a fundamental cycle basis with minimum total cost in an undirected graph. This problem is NP-hard and has several interesting applications. Since fundamental cycle bases correspond to spanning trees, we propose a local search algorithm, a tabu search and variable neighborhood search in which edge swaps are iteratively applied to a current spanning tree. We also present a mixed integer programming formulation of the problem whose linear relaxation yields tighter lower bounds than other known formulations. Computational results obtained with our algorithms are compared with those from the best available constructive heuristic on several types of graphs. This article extends the conference paper (Amaldi et al. 2004).