f—因子覆盖的图

设G为图,f是定义在V(G)上的正整数值函数。称图G的支撑子图F为f-因子如果d_(?)(x)-f(x),x∈V(G).称图G是f-因子覆盖的如果G的每条边包含在一个f-因子中.本文给出了一个图是f-因子覆盖的图的充要条件,其结果推广了C.H.C.Little et al.[1]的1-因子覆盖定理。     

循环图的1—因子化

本文证明了任意偶数阶连通循环图是1-可因子化的。 根据前两个定理这两个定理立即得证.     

与任意图正交的[0,ki]1^m—因子分解

设G是一个图,k1,…,km,是正整数,若图G的边能分解成m个边不交的[0,k1]-因子 F1,…,[0,]-l因子Fm,则称F＝{F1,…,Fm}是G 的一个[0,ki]1^m-因子分解,如果H是G的一个有m条边的了了图且对任意的1≤i≤m有E（H）E（Fi)=1,则称F与H正交,证明了若G是一个[0,k1 ,…, km-m 1]-图,。H是G的一个有m条边的子图,则图G有一个[0,ki]1^m-因子分解与H正交。     

二倍无平方因子阶的3度1-正则图

一个图叫做1-正则的, 如果它的自同构群在它的弧集上作用正则. 设n是一个无平方因子的正整数. 证明了存在2n阶3度1-正则图当且仅当n=3^tp₁p₂… p_s≥13, 其中t≤1, s≥1, p_i (1≤ i≤s)为互不相同的素数且满足3|(p_i-1). 进一步, 对每个满足上述条件的整数n, 共有2^s−1个互不同构的2n阶3度1-正则图, 并且这些图均为2n阶二面体群上的Cayley图. 由此可知, 不存在4m阶3度1-正则图, 其中m为无平方因子的奇数.     

与任意图正交的［0，ki］m1-因子分解

设G是一个图, k1,…, km是正整数.若图G的边能分解成m个边不交的［0,k1］-因子 F1,…,［0,km］-因子Fm,则称＝F1,…,Fm是G的一个［0,ki］m1-因子分解.如果H是G的一个有m条边的子图且对任意的1≤I≤m有｜E(H)∩E(Fi)｜＝1,则称与H正交.证明了若G是一个［0,k1＋…＋km－m＋1］-图,H是G的一个有m条边的子图,则图G有一个［0,ki］m1-因子分解与H正交.     

（mg＋m—1,mf—m＋1）—图的（g,f）—因子

本文证明了（ｍｇ＋ｍ－１,ｍｆ－ｍ＋１）－图具有一些特殊的（ｇ,ｆ）－因子,从而推广到了关于（ｇ,ｆ）－覆盖图和（ｇ,ｆ）－消去图的有关结果,有助于进一步研究（ｍｇ＋ｍ－１,ｍｆ－ｍ＋１）－图的正交因子分解问题。     

最小1-因子覆盖集的若干结果

假设G是一个1-可扩图．G的1-因子覆盖是G的某些1-因子的集合M使得∪M∈M M=F（G）．1-因子数目最小的1．因子覆盖称为excessive factorization．一个excessive factorization中的1．因子数目称为图G的excessive index，记为x：（G）．本文我们基于G的耳朵分解和E（C）的依赖关系给出了X＇e（G）的上界．对任意正整数k≥3，我们构造出一个图G使得A（G）=3而X＇e（G）=k．进而，我们考虑了乘积图的excessive index．     

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   

Cayley图的泛因子性质

一个κ-正则图若满足对任意正整数s,1≤s≤κ,均存在一个s-因子或一个2[s/2]因子,则称其有泛因子或偶泛因子性质.本文证明了每个奇度Cayley图是泛因子的,每个偶度Carley图是偶泛因子的.同时证明了二面体群上的每个Cayley图均是泛因子的.     

图的1-因子、f-因子和(g,f)-因子

设G是一个图且有一个1-因子F,g和f是定义在V（G）上的非负整数值函数且对每个X∈V(G)有g（X）＜f(X)≤dG（x）,且f(v(G))为偶数．（i）若对每个xy∈F有f(x)=f(y)且G－{x,y}有一个（g,f)-因子,则G有一个（g,f)-因子;（ii）若对每个xy∈F有f(X）=f（y）且G-{X,y}有f-因子,则G有f-因子.     

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.     

Excessive near 1-factorizations

We begin the study of sets of near 1-factors of graphs G of odd order whose union contains all the edges of G and determine, for a few classes of graphs, the minimum number of near 1-factors in such sets.     

Factorizations of the Complete Graph into C 5-Factors and 1-Factors

In this paper, we show that K_10n can be factored into C₅-factors and 1-factors for all non-negative integers and satisfying 2+=10n–1.Research partially supported by an NSF-AWM Mentoring Travel Grant     

Cycles in 2-Factors of Balanced Bipartite Graphs

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with . Revised: May 7, 1999     

On the size and structure of graphs with a constant number of 1-factors

We investigate the maximum number of edges in a graph with a prescribed number of 1-factors. We also examine the structure of such extremal graphs.     

On the number of 1-factors of locally finite graphs

Every infinite locally finite graph with exactly one 1-factor is at most 2-connected is shown. More generally a lower bound for the number of 1-factors in locally finite n-connected graphs is given.     

The multiplicity of 1-factors in the square of a graph

Several authors have shown that if G is a connected graph of even order then its square G² has a 1-factor. We show that the square of any connected graph of order 2n has at least n 1-factors and describe all the extremal graphs.     

PROPERTIES OF FRACTIONAL κ-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, κ]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional κ-factors is obtained.