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.     

2n-正则简单图的边连通度

图的各种连通度概念被先后用来研究网络可靠性问题.对于0到2n之间的任意一个偶数2m,构造了一个2n-正则简单图,使得其边连通度的值为2m.从而得到:2n-正则简单图的边连通度能够取{0,2,4,…,2n}中的任何一个偶数.     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

On Computing the Subset Graph of a Collection of Sets

Let a given collection of sets have size N measured by the sum of the cardinalities. Yellin and Jutla presented an algorithm which constructed the partial order induced by the subset relation (a “subset graph”) in a claimed O(N²/log N) operations over a dictionary ADT, and exhibited a collection whose subset graph had Θ(N²/log² N) edges. This paper first establishes a matching upper bound on the number of edges in a subset graph. It also presents a finer analysis of the algorithm, which confirms the claimed upper bound and shows it to be tight. A simple implementation requiring O(1) bit-parallel operations per ADT operation is presented, along with a variant of the algorithm with an implementation requiring O(N²/log N) RAM operations.     

Integer Functions on the Cycle Space and Edges of a Graph

A directed graph has a natural \mathbb Z{\mathbb {Z}} -module homomorphism from the underlying graph’s cycle space to \mathbb Z{\mathbb {Z}} where the image of an oriented cycle is the number of forward edges minus the number of backward edges. Such a homomorphism preserves the parity of the length of a cycle and the image of a cycle is bounded by the length of that cycle. Pretzel and Youngs (SIAM J. Discrete Math. 3(4):544–553, 1990) showed that any \mathbb Z{\mathbb {Z}} -module homomorphism of a graph’s cycle space to \mathbb Z{\mathbb {Z}} that satisfies these two properties for all cycles must be such a map induced from an edge direction on the graph. In this paper we will prove a generalization of this theorem and an analogue as well.     

Estimate of the Number of Edges in Subgraphs of a Johnson Graph

Doklady Mathematics - New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained.     

On the Number of Subgraphs of a Specified Form Embedded in a Random Graph

Let U ₁, U ₂,... be a sequence of i.i.d. random elements in R^d. For x>0, a graph G _n (x) may be formed by connecting with an edge each pair of points in that are separated by a distance no greater than x. The points of G _n (x) could represent the stations in a telecommunications network and the edge set the lines of communication that exist among them. Let be a collection of graphs on mn points having a specified form or structure, and let denote the number of subgraphs embedded in G _n (x) and contained in . It is shown that a SLLN, CLT and LIL for follow easily from the theory of U-statistics. In addition, a uniform (in x) SLLN is proved for collections that satisfy a certain monotonicity condition. Some applications are mentioned and the results of some simulations presented. The scaling constants appearing in the CLT are usually hard to obtain. These are worked out for some special cases.     

Estimate of the Number of Edges in Special Subgraphs of a Distance Graph

Mathematical Notes - The classical problem of estimating the number of edges in a subgraph of a special distance graph is considered. Old results are significantly improved.     

3-连通3-正则图生成树外的可去边

G是3-连通图，e是G中的一条边．若G-e是3-连通图的一个剖分，则称e是3-连通图的可去边．否则，e是G中不可去边．本给出3-连通3-正则图中生成树外可去边的分布情况及数目．     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

The Size of a Maximum Subgraph of the Random Graph with a Given Number of Edges

Doklady Mathematics - We have proven that the maximum size k of an induced subgraph of the binomial random graph $$G(n,p)$$ with a given number of edges $$e(k)$$ (under certain conditions on this...     

Maximum Number of Contractible Edges on Hamiltonian Cycles of a 3-Connected Graph

 We show that if G is a 3-connected hamiltonian graph of order at least 5, then there exists a hamiltonian cycle C of G such that the number of contractible edges of G which are on C is greater than or equal to . Received: July 31, 2000 Final version received: December 12, 2000 Acknowledgments. I would like to thank Professor Yoshimi Egawa for the help he gave to me during the preparation of this paper.     

最大边数的Cordial图的构造

对于n阶Cordial图G，本给出G的边数的上确界e^*，并给出边数达到e^*的Cordial图的构造。     

The First Three Levels of an Order Preserving Hamiltonian Path in the Subset Lattice

Consider the lattice whose elements are the subsets of the set of positive integers not greater than n ordered by inclusion. The Hasse diagram of this lattice is isomorphic to the n-dimensional hypercube. It is trivial that this graph is Hamiltonian. Let be a Hamiltonian path. We say it is monotone, if for every i, either (a) all subsets of S _i appear among S ₁,...,S _i − 1, or (b) only one (say S) does not, furthermore S _i + 1 = S. Trotter conjectured that if n is sufficiently large, then there are no monotone Hamiltonian paths in the n-cube. He also made a stronger conjecture that states that there is no path with the monotone property that covers all the sets of size at most three. In this paper we disprove this strong conjecture by explicitly constructing a monotone path covering all the 3-sets.     

k连通图可缩边导出子图的连通性

引进S1 3边形的概念 .证明了 ,对于k(k =3或 4)连通图G ,若G无S1 3边形 ,则 是 2连通的 ;另外也得到 ,设G是k(k≥ 2 )连通图 ,若对G的任一断片F ,有｜F｜ >[k/2 ]+ 1 ,则 是 2连通的 .从而改进并推广了N .Dean的结论 .     

最大度为8不含特定子图的平面图的全染色

全染色是对图G的顶点和边同时进行正常染色,至少要用Δ+1个色才能对图G进行正常全染色.本文运用权转移的方法,证明了最大度为8的不含特定子图的简单平面图是9-全可染的.     

Constructing the Spectrum of Packings and Coverings for the Complete Graph with Stars with up to Five Edges

The packing and covering problems have been considered for several classes of graphs. For instance, Bryant et. al. have investigated the packing problem for paths and cycles, and the packing and covering problems for 3-cubes. The packing and covering problems were settled for stars with up to six edges by Roditty. In this paper, for every possible leave graph (excess graph), we find a corresponding maximum packing (minimum covering) of the complete graph with stars with up to five edges.     

图的倍图与补倍图

计算机科学数据库的关系中遇到了可归为倍图或补倍图的参数和哈密顿圈的问题．对简单图C,如果V（D（G））：V（G）∪V（G′）E（D（G））=E（C）∪E（C″）U{vivj′｜vi∈V（G）,Vj′∈V（G′）且vivj∈E（G））那么,称D（C）是C的倍图,如果V（D（G））=V（C）∪V（G′）,E（D（C））：E（C）∪E（G′）∪{vivj′}vi∈V（G）,vj′∈V（G’）and vivj∈（G））,称D（C）是G的补倍图,这里G′是G的拷贝．本文研究了D（G）和D的色数,边色数,欧拉性,哈密顿性和提出了D（G） 的边色数是D（G）的最大度等公开问题．