图的孤立断裂度

连通图G的孤立断裂度isc(G)=max{i(G-S)-|S|:S∈C(G)},其中i(G-S)是G-S中的孤立点数,C(G)是G的点割集.本文研究了孤立断裂度和图的其它一些参数的关系.讨论了孤立断裂度取特殊值的一些图,证明了圈、连通二部图、连通二部图的联图以及树和圈的补图的孤立断裂度都达到最小.给出了具有给定阶数和最大度的村的最大、最小孤立断裂度.     

Bicliques in Graphs I: Bounds on Their Number

Bicliques are inclusion-maximal induced complete bipartite subgraphs in graphs. Upper bounds on the number of bicliques in bipartite graphs and general graphs are given. Then those classes of graphs where the number of bicliques is polynomial in the vertex number are characterized, provided the class is closed under induced subgraphs. Received January 27, 1997     

Bi-complement Reducible Graphs

We introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG = (W ∪ B, E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphofGis the graph having the same vertex setW ∪ BasG, while its edge set is equal toW × B − E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs.     

Chaotic Numbers of Complete Bipartite Graphs and Tripartite Graphs

In this paper, we study the chaotic numbers of complete bipartite graphs and complete tripartite graphs. For the complete bipartite graphs, we find closed-form formulas of the chaotic numbers and characterize all chaotic mappings. For the complete tripartite graphs, we develop an algorithm running in O(n ⁴ ₃) time to find the chaotic numbers, with n ₃ the number of vertices in the largest partite set.Research supported by NSC 90-2115-M-036-003.The author thanks the authors of Ref. 6, since his work was motivated by their work. Also, the author thanks the referees for helpful comments which made the paper more readable.     

一些图的邻点可区别关联着色

在图的关联着色概念的基础上定义了图的邻点可区别关联着色及邻点可区别关联色数,研究了圈、完全二部图、Cm.Fn图的邻点可区别关联着色,并确定了它们的邻点可区别关联色数.     

Perfect Elimination and Chordal Bipartite Graphs

We define two types of bipartite graphs, chordal bipartite graphs and perfect elimination bipartite graphs, and prove theorems analogous to those of Dirac and Rose for chordal graphs (rigid circuit graphs, triangulated graphs). Our results are applicable to Gaussian elimination on sparse matrices where a sequence of pivots preserving zeros is sought. Our work removes the constraint imposed by Haskins and Rose that pivots must be along the main diagonal.     

二部图的单特征值

一个图的特征值通常指的是它的邻接矩阵的特征值,在图的所有特征值中,重数为1的特征值即所谓的单特征值具有特殊的重要性.确定一个图的单特征值是一个比较困难的问题,主要是没有一个通用的方法.1969年,Petersdorf和Sachs给出了点传递图单特征值的取值范围,但是对于具体的点传递图还需要根据图本身的特性来确定它的单特征值.给出一类正则二部图,它们是二面体群的凯莱图,这类图的单特征值中除了它的正、负度数之外还有0或者±1,而它们恰好是Petersdorf和Sachs所给出的单特征值范围内的中间取值.     

图的{P_4}—分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上.记Pl长度为l-1的路,如果G能够分解成若干个Pl,则称G存在{Pl}—分解.关于图的给定长路分解问题主要结果有:(i)连通图G存在{P3}—分解当且仅当G有偶数条边(见[1]);(ii)连通图G存在{P3,P4}—分解当且仅当G不是C3和奇树,这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树(见[3]).本文讨论了3正则图的{P4}—分解情况,并构造证明了边数为3k(k热∈Z且k≥2)的完全图Kn和完全二部图Kr,s存在{P4}—分解.     

Chordal Bipartite Graphs with High Boxicity

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.     

Graphs with the maximum or minimum number of 1-factors

Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs which are unions of complete regular bipartite graphs, and complete graphs with a matching removed. The first family is determined using the Alon and Friedland bound. For the second family, we show that a graph transformation which is known to increase network reliability also increases the number of 1-factors. In fact, more is true: this graph transformation increases the number of k-factors for all k≥1, and “in reverse” also shows that in general, threshold graphs have the fewest k-factors. We are then able to determine precisely which threshold graphs have the fewest 1-factors. We conjecture that the same graphs have the fewest k-factors for all k≥2 as well.     

On the Jump Number Problem in Hereditary Classes of Bipartite Graphs

We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying this condition, we propose polynomial algorithms to solve the jump number problem.     

图的距离不大于β的任意两点可区别的边染色

本文提出了图的距离不大于β的任意两点可区别的边染色,即D(β)-点可区别的边染色(简记为D(β)-VDPEC)．并得到了一些特殊图类,如圈、完全图、完全二部图、扇、轮、树以及一些联图的D(β)-点可区别的边色数,文后提出了相关的猜想．     

Linear Balanceable and Subcubic Balanceable Graphs*

In Math Program 55(1992), 129–168, Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove that cubic balanced graphs contain a pair of twins, a result that was conjectured by Morris, Spiga, and Webb in ( Discrete Math 310(2010), 3228–3235).     

关于一类二部图的均匀邻点可区别全染色

若图的邻点可区别全染色的各色所染元素数之差不超过1,则称该染色法为图的均匀邻点可区别全染色,而所用的最少颜色数称为该图的均匀邻点可区别全色数.本文给出了一类二部图的均匀邻点可区别全染色数.     

Enumerative Properties of Ferrers Graphs

We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.     

给定染色数的无符号Laplace谱半径

设Gkn(k≥2)为n阶的染色数为k的连通图的集合.本文确定了Gkn中具有极大无符号Laplace谱半径的图,即k=2时为完全二部图,k≥3时为Turn图.本文也讨论了Gkn中的具有极小无符号Laplace谱半径的图,对k≤3的情形给出了此类图的刻画.     

Nowhere‐Zero Flows on Signed Complete and Complete Bipartite Graphs

Bouchet's conjecture asserts that each signed graph which admits a nowhere‐zero flow has a nowhere‐zero 6‐flow. We verify this conjecture for two basic classes of signed graphs—signed complete and signed complete bipartite graphs by proving that each such flow‐admissible graph admits a nowhere‐zero 4‐flow and we characterise those which have a nowhere‐zero 2‐flow and a nowhere‐zero 3‐flow.     

偶匹配可扩性的极图问题(英文)

设G是含有完美匹配的简单图.称图G是偶匹配可扩的(BM-可扩的),如果G的每一个导出子图是偶图的匹配M都可以扩充为一个完美匹配.极图问题是图论的核心问题之一.本文将刻画极大偶匹配不可扩图,偶图图类和完全多部图图类中的极大偶匹配可扩图.     

Minimum Size of n-Factor-Critical Graphs and k-Extendable Graphs

We determine the minimum size of n-factor-critical graphs and that of k-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of k-extendable non-bipartite graphs for k = 1, 2, and pose a related conjecture for general k.     

Neighbourhood-Perfect Line Graphs

We show that the line graph of any balanced hypergraph is neighbourhood-perfect. This result is a hypergraphic extension of a recent theorem in [GKLM] saying that the line graphs of bipartite graphs are neighbourhood-perfect. The note contains also a graphical extension of the same theorem: the characterization of all graphs with neighbourhood-perfect line graph. The proof relies on a characterization of neighbourhood-perfect graphs among line graphs in terms of forbidden induced subgraphs.