无圈超图的计数

研究了标号超图的计数, 得到2个公式: 一个是关于严格(D)-连通无圈齐超图的显式计数公式, 另一个是关于线性无圈超图数目的递推公式.     

超图的路和圈

超图是离散数学中最一般最复杂的结构 .无圈超图已被证明在数据库设计中非常有用 .从关系数据的结构出发 ,建立了关于超图的路、连通性和圈的新的公理系统 .该系统与特殊情形———图是符合的 .引入了虚圈和实圈的概念 ,这是一对相关联的概念 .虚圈在特殊情形———图中不存在 ,退化掉了 .定义了超图圈的相关性和独立性 ,给出了超图中最大独立实圈数目的计数公式 ,对特殊情形———图 ,这个公式就是Euler公式 .     

超图中的C-圈

在本文我们给出了一个新的定义C-圈.设f(n,k,r)是不含C-圈的n阶r-一致超图的最大可能边数,我们主要是确定f(n,k,r)或给出它的一个下界.另外,我们给出了超图不含C-圈的一个充分必要条件.     

完全3-一致超图的圈分解

The problem of decomposing a complete 3-uniform hypergraph into Hamilton cycles was introduced by Bailey and Stevens using a generalization of Hamiltonian chain to uniform hypergraphs by Katona and Kierstead. Decomposing the complete 3-uniform hypergraphs K_n~(3) into k-cycles(3 ≤ k n) was then considered by Meszka and Rosa. This study investigates this problem using a difference pattern of combinatorics and shows that K_(n·5m)~(3) can be decomposed into 5-cycles for n ∈{5, 7, 10, 11, 16, 17, 20, 22, 26} using computer programming.     

匀称无圈超图的计数

研究了标号匀称无圈超图的计数, 得到了一般的$n$阶标号r-匀称(d)-森林和n阶标号r-匀称(d)-真森林的递推公式,并分别得到了包含和不包含独立点的$n$阶标号森林的计数显式.     

严格(d)-连通无圈超图的计数

研究了一般的标号严格(d)-连通无圈超图的计数,得到了n阶标号严格(d)-连通无圈超图的计数公式.     

关于不含k-C-圈的n阶γ-一致超图的若干结果

主要讨论了不含k-C-圈的n阶γ-一致超图,对不同的k, 分别得出了它的极大边数的一个下界,并且得出在有些情况下它的下界是最大的.另外,我们得到了Krn含k-C-圈的一个充分必要条件.     

关于不含k-C-圈的n阶r-一致超图的若干结果

主要讨论了不含k-C-圈的n阶r-一致超图,对不同的k,分别得出了它的极大边数的一个下界,并且得出在有些情况下它的下界是最大的.另外,我们得到了Krn含k-C-圈的一个充分必要条件.     

无标号真严格(d)-连通无圈超图的计数

本文得到了无标号真严格(d)-连通无圈超图的计数公式,并得到了无标号真严格(d)-连通同胚k不可约无圈超图的计数公式.     

3—连通正则无爪图的Hamilton圈

本文证明了：任一阶数不超过６ｋ－４的３－连通ｋ－正则无爪图是Ｈａｍｉｔｏｎ的。     

Enumeration of Maximum Acyclic Hypergraphs

Abstract Acyclic hypergraphs are analogues of forests in graphs.They are very useful in the design ofdatabases. In this article,the maximum size of an acvclic hypergraph is determined and the number of maximumγ-uniform acyclic hypergraphs of order n is shown to be (_(r-1)~n)(n(r-1)-r~2 2r)~(n-r-1).     

On Acyclic Hypergraphs of Minimal Prime Models

We discuss the question of restoring the structural properties of theories from the hypergraphs of minimal prime models. We describe the spectrum and the main model-theoretic properties of acyclic complete theories with the property of extension of isomorphisms of families of minimal prime models.     

Covering Complete Hypergraphs with Cuts of Minimum Total Size

A cut [X, V − X] in a hypergraph with vertex-set V is the set of all edges that meet both X and V − X. Let s _r (n) denote the minimum total size of any cover of the edges of the complete r-uniform hypergraph on n vertices K_n^r{K_n^r} by cuts. We show that there is a number n _r such that for every n > n _r , s _r (n) is uniquely achieved by a cover with ?\fracn-1r-1?{\lfloor \frac{n-1}{r-1}\rfloor} cuts [X _i , V − X _i ] such that the X _i are pairwise disjoint sets of size at most r − 1. We show that c₁r2^r < n_r < c₂r⁵2^r{c_1r2^r < n_r < c_2r^52^r} for some positive absolute constants c ₁ and c ₂. Using known results for s ₂(n) we also determine s ₃(n) exactly for all n.     

Lower Bounds on the Size of Maximum Independent Sets and Matchings in Hypergraphs of Rank Three

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let and be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3‐uniform hypergraph of order at least 8 and with maximum degree at most three, then and there is a connected 3‐uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on vertices, where denotes the maximum degree in H, then and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory     

超图中的着色问题

本文是近三十年来有关超图中涉及的着色问题的综述。它包含了有关超图着色中的基本结果,临界可着色性,２－可着色性,非２－可着色性以及在超图中与顶点着色、边着色和其它着色相关的极值问题。     

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.     

On Equitable Colorings of Hypergraphs

Mathematical Notes - A two-coloring is said to be equitable if, on the one hand, there are no monochromatic edges (the coloring is regular) and, on the other hand, the cardinalities of color...     

A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C _m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C _m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r ₁,r ₂,…,r _l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r ₁,r ₂,? ,r _l }-hypergraphs and {r ₁,r ₂,? ,r _l }-hypergraphs concerning the second question.