二分图中关于Enomoto问题的结果

设k,n1和n2是3个正整数,G=（V1,V2;E）是一个二分图,使得｜V1｜=n1,｜V2｜=n2,其中n1≥2k＋1,n2≥2k＋1并且n1-n2≤1．如果对任意不相邻的x∈V1和y∈V2,都有d（x）＋d（y）≥2k＋2,则G包含k个相互独立的圈．以上结果部分地回答了Enomoto提出的关于二分图有独立圈的问题．     

二分图中度条件和k-因子的存在性

本文主要研究了二分图中任意一对距离为２的顶点的度数与ｋ－因子关系,给出了二分图有ｋ因子的若干充分条件,并说明这些条件是最好的可能,从而证明了Ｎｉｓｈｉｍｕｒａ提出的问题对二分图成立。     

关于图因子的邻集条件

设F为图G的一个支撑子图.如果对所有x∈V(G),有d_F(x)∈{1,3,…,2n-1),则称F为G的一个(1,3,…,2n-1)一因子;如果对所有x∈V(G),有d_F(x)=k,则称F为G的一个k-因子.本文以图的顶点邻集对一个图具有包含任一条给定边的{1,3,…,2n-1)-因子和k-因子分别给出了充分条件.     

关于k-消去二分图的一些结果

图 G的一个 k-正则支撑子图称为 G的 k-因子 ,若对 G的任一边 e,图 G- e总存在一个 k-因子 ,则称 G是 k-消去图 .证明了二分图 G=( X,Y) ,且 | X | =| Y|是 k-消去图的充分必要条件是 k| S|≤ r1 + 2 r2 +…+ k( rk+… + rΔ) - ε( S)对所有 S X成立 .并由此给出二分图是 k-消去图的充分度条件 .     

极大非独立集可削去的因子临界图

如果对一个简单图G的每一个与G的顶点数同奇偶的独立集I，都有G-I有完美匹配，则称G是独立集可削去的因子临界图．如果图G不是独立集可削去的因子临界图，而对任意两个小相邻的顶点x与y，G xy足独立集可削去的因子临界图，则称G足极大非独立集可削去的因子临界图，本刻画了极大非独立集可削去的因子临界图。     

图的最大亏格与2-因子

图Ｇ的一个２因子Ｆ就是Ｇ的这样一个支撑子图，使其任何节点ｖ∈Ｖ的次ｄＦ（ｖ）＝２．易见，Ｇ的每个２因子均为无公共节点的圈之并．若Ｆ的每个圈的长均为３（或４），则称Ｇ含有一个三角形（或四边形）２因子．Ｍ．ｋ∨ｏｖｉｅｒａ［５］得到了含有三角形２因子的３－正则图的最大亏格．本文在３－正则图上，引进了扩张运算和讨论了与最大亏格和Ｂｅｔｉ亏数之间的关系．利用这些运算，得到了所有含四边形２因子的连通３－正则图是上可嵌入的，即γＭ（Ｇ）＝ｎ４（ｎ为Ｇ的节点数ｎ＝｜Ｖ（Ｇ）｜）．然后，基于此证明了含四边形２因子且所有节点ｖ∈Ｖ的次ｄＧ（ｖ）＝３（ｍｏｄ４）的图Ｇ均为上可嵌入的     

一类二分图的k-匹配数

设G是一个具有二分类(X_1,X_2)的简单偶图,|X_1|=|X_2|=n,如果对于给定的c>0,|M(S)|≥(1+c)|S|对任意满足|S|≤n/2的S(?)X_i(i=1,2)都成立,其中N(S)是S的邻集,则称G是(n,c)-扩张图.给出了(n,c)-扩张图的k-匹配数与完美匹配数之比的顺从界.     

图中的最大分数(0,f)-因子

本语文给出了图的一个（０,ｆ）因子是最大因子的特征,并得到了一个图有（ｇ,ｆ）－因子的充分条件,从而了关于分数对集和１－因子的有关结果。     

均衡二部图中含2k条指定边的k个独立圈及2-因子

得到了对于二部图G=(V_1,V_2;E),当|V_1|=|V_2|=n≥2k+1时的结果:对G中任意2k条独立边e_1,e_1~*,…,e_k,e_k~*,G中一定存在k个独立的4-圈C_1,C_2,…,C_k,使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i).并在此基础上进一步证明了当|V_1|=|V_2|=n≥3k时若对任意两顶点x∈V_1,y∈V_2,都有d(x)+d(y)≥2n-k+1成立,则G有一个2-因子含有k+1个独立圈C_1,C_2,…,C_(k+1)使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i)且|C_i|=4.     

具有唯一完美匹配的D-图

为了研究具有完美匹配图的Tuttc集和极端集,文献[1,2]提出了一种新的图运算,并且得到了许多有趣的性质。本文中,我们刻画了level（G）=0的具有唯一完美匹配的饱和图G,并且确定了具有唯一完美匹配图的D-图的边数的紧上界。     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

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.     

3-正则图的不共边的完美匹配（英文）

研究3-正则图的一个有意义的问题是它是否存在k个没有共边的完美匹配.关于这个问题有一个著名的Fan-Raspaud猜想：每一个无割边的3-正则图都有3个没有共边的完美匹配.但这个猜想至今仍未解决.设dim（P（G））表示图G的完美匹配多面体的维数.本文证明了对于无割边的3-正则图G,如果dim（P（G））≤14,那么k≤4：如果dim（P（G））≤20,那么k≤5.     

二部图的E-H-不可收缩性

By End（G） and hEnd（G） we denote the set of endomorphisms and half-strong endomorphisms of a graph G respectively. A graph G is said to be E-H-unretractive if End（G） = hEnd（G）. A general characterization of an E-H-unretractive graph seems to be difficult. In this paper, bipartite graphs with E-H-unretractivity are characterized explicitly.     

Bipartite Subgraphs of Graphs with Maximum Degree Three

We prove that for a connected graph G with maximum degree 3 there exists a bipartite subgraph of G containing almost of the edges of G. Furthermore, we completely characterize the set of all extremal graphs, i.e. all connected graphs G=(V, E) with maximum degree 3 for which no bipartite subgraph has more than of the edges; |E| denotes the cardinality of E. For 2-edge-connected graphs there are two kinds of extremal graphs which realize the lower bound . Received: July 17, 1995 / Revised: April 5, 1996     

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...     

Group Weighted Matchings in Bipartite Graphs

Let G be a bipartite graph with bicoloration {A, B}, |A| = |B|, and let w : E(G) K where K is a finite abelian group with k elements. For a subset S E(G) let . A Perfect matching M E(G) is a w-matching if w(M) = 1.A characterization is given for all w's for which every perfect matching is a w-matching.It is shown that if G = K _k+1,k+1 then either G has no w-matchings or it has at least 2 w-matchings.If K is the group of order 2 and deg(a) d for all a A, then either G has no w-matchings, or G has at least (d – 1)! w-matchings.R. Meshulam: Research supported by a Technion V.P.R. Grant No. 100-854.     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.