循环图C_(2n)(1,(2n+1)/3)的匹配可扩性

称图G是k-偶匹配可扩的,是指G的每一个基数不大于k(1≤k≤(|V(G)|-2)/2)的偶匹配M都可以扩充为G的一个完美匹配.根据循环图的性质研究了图C_(2n)(1,(2n+1)/3)的匹配可扩性,证明了对于任意的n(n≥4),C_(2n)(1,(2n+1)/3)是3-偶匹配可扩的.     

匹配可扩图的结构

设G是一个有限的简单连通图。D（G）表示V（G）的一个子集,它的每一个点至少有一个最大匹配不覆盖它。A（G）表示V（G）-D（G）的一个子集,它的每一个点至少和D（G）的一个点相邻。最后设C（G）=V（G）-A（G）-D（G）。在这篇章中,下面的被获得。⑴设u∈V（G）。若n≥1和G是n-可扩的,则（a)C(G-u)=φ和A（G-u)∪{u}是一个独立集,（b)G的每个完美匹配包含D（G-u)的每个分支的一个几乎守美匹配,并且它匹配A（G-u)∪{u}的所有点与D（G-4）的不同分支的点。⑵若G是2-可扩的,则对于u∈V（G）,A（G-u）∪{u}是G的一个最大障碍且G的最大障碍的个数是2或是│V(G)│.⑶设X=Cay(Q,S),则对于u∈Q,(a)A(X-u)=φ=C（G-u)和X-u是一个因子临界图,或（b)C(X-u)=φ和X的两部是A（X-u)∪{u}和D（X-u)且│A(X-u)∪{u}│=│D(X-u）│。⑷设X=Cay(Q,S),则对于u∈Q,A（X-u)∪{u}是X的一个最大障碍且X的最大障碍的个数是2或是│Q│。     

奇图的匹配可扩性

设G是一个图,n,k和d是三个非负整数,满足n+2k+d≤|V(G)|-2,|V(G)|和n+d有相同的奇偶性.如果删去G中任意n个点后所得的图有k-匹配,并且任一k-匹配都可以扩充为一个亏d-匹配,那么称G是一个(n,k,d)-图.Liu和Yu[1]首先引入了(n,k,d)-图的概念,并且给出了(n,k,d)-图的一个刻划和若干性质. (0,k,1)-图也称为几乎k-可扩图.在本文中,作者改进了(n,k,d)-图的刻划,并给出了几乎k-可扩图和几乎k-可扩二部图的刻划,进而研究了几乎k-可扩图与n-因子临界图之间的关系.     

无爪图的导出匹配可扩性

若图G的一个匹配M也是G的点导出子图,则称M是图G的一个导出匹配．我们称图G是导出匹配可扩的,若它的任何一个导出匹配可以扩充成一个完美匹配,本文我们讨论无爪图的导出匹配可扩性,得出如下结论,并同时指出这些结果是最好可能的．设图G是有2n个顶点的无爪图,1．若图G是最小度大于或等于2 1,则图G是导出匹配可扩的．2．若图G是局部2连通的,则留G是导出匹配可扩的．3．若图G是k正则的且k≥n,则图G是导出匹配可扩的．     

I-图的可扩性

如果连通图I中任意n条点不交的边都包含在一个完美匹配中,就称I是n-可扩的.证明了真I图I(n,j,k)是1-可扩的;当n≠3j或者3k时,真I-图是2-可扩的.     

导出匹配,极大导出匹配可扩图和导出匹配数

目前我们已知的极大导出匹配可扩图只有Kn,n和K2n.为了研究它们是否是仅有的极大导出匹配可扩图,我们考虑了匹配数,导出匹配数,极大导出匹配可扩图以及一个相关的猜想,并得出了若干相关的结果.     

导出匹配问题的NP-完全性以及导出匹配可扩问题的CO-NP-完全性

图Ｇ的一个匹配Ｍ是导出的,若Ｍ是图Ｇ的一个导出子图。图Ｇ是导邮匹配可扩的（简记ＩＭ－可扩的）,若图Ｇ的任一导出匹配均含于图Ｇ的一个完美匹配当中。本文我们将证明如下结果。⑴对无爪图而言,问题“给定图Ｇ以及一个正整数ｒ,确定是否存在图Ｇ的一个导出匹配Ｍ使得Ｍ≥ｒ”是ＮＰ－完全的。⑵对直径为２的图以及直径为３的偶图,问题“确定一个给定图是否为导出匹配可扩的”是ＣＯ－ＮＰ完全的;而对完全多部图而言,问题“     

导出匹配可扩图的局部运算

称图G为导出匹配图可扩的（简称为IM－可扩的），如果图G的一每个导出匹配都包含在G的一个完美匹配中，本给出了导出匹配可扩图的一些局部运算。     

关于n-可扩图的一些结果

本文讨论了关于m-可扩图的两个极值问题；并考查了下述图类的n-可扩性；正则偶图，单位区间图和分裂图。     

偶匹配可扩性的极图问题

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.     

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.     

二分图中含有完美对集的2因子

该文证明若G是2n阶均衡二分图，δ(G)≥(2n-1)/3，则对任何正整数k，n≥4k时，任给G的一个完美对集M，G中存在一个包含M的所有边的恰含k个分支的2 因子(k=1，n=5且δ(G)=3除外）. 特别k=2时，在条件n≥5且δ(G)≥(n+2)/2下，结论也成立. 这里所给的δ(G)的下界是最好的可能.      

3阶幂图的广义导出匹配可扩性

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.     

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...     

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.     

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.     

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.     

Accessibility and Reliability for Connected Bipartite Graphs

In the recently published atlas of graphs [9] the general listing of graphs with diagrams went up to V=7 vertices but the special listing for connected bipartite graphs carried further up to V=8. In this paper we wish to study the random accessibility of these connected bipartite graphs by means of random walks on the graphs using the degree of the gratis starting point as a weighting factor. The accessibility is then related to the concept of reliability of the graphs with only edge failures. Exact expectation results for accessibility are given for any complete connected bipartite graph N₁ cbp N₂ (where cbp denotes connected bipartite) for several values of J (the number of new vertices searched for). The main conjecture in this paper is that for any complete connected bipartite graph N₁ cbp N₂: if |N₁–N₂| 1, then the graph is both uniformly optimal in reliability and optimal in random accessibility within its family. Numerical results are provided to support the conjecture.     

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