3-一致超图的反馈数研究

超图H=(V,E)顶点集为V,边集为E.S(C)V是H的顶点子集,如果HS不含有圈,则称S是H的点反馈数,记tc(H)是H的最小点反馈数.本文证明了:(i)如果H是线性3-一致超图,边数为m,则tc(H)≤m/3;(ii)如果日是3-一致超图,边数为m,则Tc(H)≤m/2并且等式成立当且仅当日任何一个连通分支是孤立...     

一致超图中Berge线性森林的Turán数

设F是一个图,■是一个超图,如果存在一个双射φ:E(F)→E(■),使得?e∈E(F)有e?φ(e),那么称超图■是Berge-F.不含Berge-F作为子超图的n阶r-一致超图所能达到的最大边数称为Berge-F的Turán数,记作ex_r(n,Berge-F).线性森林是指连通分支全是路或者孤立顶点的图.设■_n,k是一类含有n个顶点k条边的线性森林图族.本文研究了r-一致超图中Berge-■_n,k的Turán数.当r≥k+1和3≤r≤■(k-1)/2■-1时,分别确定了ex_r(n,Berge-■_n,k)的精确值;当■(k-1)/2■≤r≤k时,给出了ex_r(n,Berge-■_n,k)的上界.     

强二部超图的饱和数

给定r-图F,称一个r-图G是F-饱和的,如果G不包含F,但是对于每条满足e∈E((G))的r-边e,G添加该边后会包含F,其中(G)表示G的补图.r-图F的饱和数,记为satr(n,F),指的是n个顶点的F-饱和r-图的最小边数.令Srl,m为一个有l+m个顶点的r-图,其边集合由所有与某固定l-集合交集非空的边组成...     

关于D-完全一致混合超图上色数的一个结论的推广

混合超图的上、下色数的研究是超图研究中一个重要的话题.由于超图本身结构上的复杂性,近年来对超图色性的研究也近局限于对一些特殊图类的研究,其中完全一致混合超图是最为热门的图类之一.给出了D完全（C不完全）一致混合超图的概念,并运用组合数学中有关分划的思想和方法对该图类的色性进行了进一步的研究,对相关文献中给出的结论进行了推广,得到了一个较为一般化的结论.并在该定理的证明中得到并证明了一个关于混合超图C稳定集的重要论断,对超图色性研究有着重要的意义.     

3一致 C-超图的最小边数

混合超图是含有两类超边的超图,一类称为C-超边,一类称为D-超边,它们的区别主要体现在染色要求上．混合超图的染色,要求每一C-超边至少有两个点染相同的颜色,而每一D-超边至少有两个点染不同的颜色．所用的最大颜色数称为对应混合超图的上色数,所用的最小颜色数称为对应混合超图的下色数．上、下色数与边数有密切关系．作者在文献[2]中证明了具有最小上色数的3一致C-超图边数的一个下界为‘n(n-2)/3’,其中n为对应混合超图的顶点数．该文证明当n=2k 1时,该下界是可以达到的．     

4一致L—超图的最小边数的上界

主要讨论了4一致L—超图的最小边数与最小上色数的关系，给出了上色数为3的4一致L—超图的最小边数的一个上界。     

混合超图的染色理论

混合超图是含有两种超边的超图，一种称为D-超边，一种称为C-超边，它们的区别主要体现在染色要求上．混合超图的染色，要求每-D-超边至少有两个点染不同的颜色，每一C-超边至少有两个点染相同的颜色．用颜色最多的染色所用的颜色数称为该混合超图的上色数，用颜色最少的染色所用的颜色数称为该混合超图的下色数．混合超图的染色理论是目前国际组合学界比较新的研究课题之一．本文主要概括介绍关于混合超图染色理论已经取得的一些成果，其中包含本文作者的研究成果．并提出了一些可供进一步研究的问题．     

4一致C-超图的最小边数的上界(英文)

主要讨论了 4一致C 超图的最小边数与最小上色数的关系 ,给出了上色数为 3的 4一致C 超图的最小边数的一个上界 .     

本原简单超图的指数集

设H是一个超图, 用H+*和L(H)分别表示H的对偶超图和线图. 定义H的邻接图是由L(H+*)和H的所有环组成的图, 记作G-H. 若G-H是本原的, 则称H是本原的, 并称γ(G-H)为H的指数. 该文得到了所有n阶本原简单超图以及所有秩不小于3的n阶本原简单超图的指数集, 并分别刻划了其极超图.     

r-一致超图Ramsey函数的渐近下界(英文)

本文利用Lovasz局部引理的Spencer形式和对称形式给出r-一致超图Ramsey函数的渐近下界.证明了:对于任意取定的正整数f0,使得当n→∞时,有R~((r))(m~l,n~(k-l))≥(c-o(1))(n~(r-1)/logn)~■.特别地,R~((r))_k(n)≥(1-o(1))n/e k~■(n→∞).对于任意取定的正整数s≥r+1和常数δ>0,α≥0,如果F表示阶为s的r-一致超图,■表示阶为t的r-一致超图,且■的边数满足m(■)≥(δ-o(1))t~r/(logt)α(t→∞),则存在c=c(s,δ,α)>0,使得R~((r))(F,■)≥(c-o(1))(t~(r-1)/(logt)~l+(r-l)α)~(m(F)-l/s-r).     

On the Strong Chromatic Number of Random Hypergraphs

Doklady Mathematics - We study the probability threshold for the property of strong colorability with a given number of colors of a random $$k$$ -uniform hypergraph in the binomial model...     

The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels

The notion of the star chromatic number of a graph is a generalization of the chromatic number. In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. The first two families are derived from a 3-or 5-wheel by subdivisions, their star chromatic numbers being 2+2/(2n + 1), 2+3/(3n + 1), and 2+3(3n−1), respectively. The third family of planar graphs are derived from n odd wheels by Hajos construction with star chromatic numbers 3 + 1/n, which is a generalization of one result of Gao et al. Received September 21, 1998, Accepted April 9, 2001.     

The Chromatic Spectrum of Mixed Hypergraphs

 A mixed hypergraph is a triple ℋ=(X, ?, ?), where X is the vertex set, and each of ?, ? is a list of subsets of X. A strict k-coloring of ℋ is a surjection c:X→{1,…,k} such that each member of ? has two vertices assigned a common value and each member of ? has two vertices assigned distinct values. The feasible set of H is {k: H has a strict k-coloring}. Among other results, we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number 1 or is an interval starting with 1. For the set {s,t} with 2≤s≤t−2, the smallest realization has 2t−s vertices. When every member of ?∪? is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers. Received: May 24, 1999 Final version received: August 31, 2000     

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least \(c \left( {\begin{array}{c}|V(H)|\\ r-1\end{array}}\right) \) has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomassé and then solved by Allen, Böttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turán density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobás–Erd?s graphs defined earlier by the authors.     

图的星色数

给出了一些星色数为４的平面图,它们不含有轮图作为子图,这回答了Ｚｈｕ的一个问题,给出了一类４连通平面图其星色数在３与４之间,这也回答了Ａｂｂｏｔｔ和Ｚｈｏｕ的一个问题,应用图的同态概念,讨论了某些图的字典积的星色数,证明了一个图及其补图的星色数的和与积所满足的两个不等式。     

Uniform Mixed Hypergraphs: The Possible Numbers of Colors

For a mixed hypergraph , where and are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’ such that every meets some class in more than one vertex, and every has a nonempty intersection with at least two classes. The feasible set of , denoted , is the set of integers k such that admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309–318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either or S is an ‘interval’ for some integer k ≥ 1. In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all and all , r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either for some natural number k ≥ r − 1, or S is of the form where S′′ is any (possibly empty) subset of and S′ is either the empty set or {r − 1} or an ‘interval’ {k, k + 1, ..., r − 1} for some k, 2 ≤ k ≤ r − 2. We also prove that all these feasible sets can be obtained under the restriction , i.e. within the class of ‘bi-hypergraphs’. Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613.     

Competition Numbers and Phytogeny Numbers of Connected Graphs and Hypergraphs

Let D be a digraph.The competition graph of D is the graph having the same vertex set with D and having an edge joining two different vertices if and only if they have at least one common out-neighbor in D.The phylogeny graph of D is the competition graph of the digraph constructed from D by adding loops at all vertices.The competition/phylogeny number of a graph is the least number of vertices to be added to make the graph a competition/phylogeny graph of an acyclic digraph.In this paper,we show that for any integer k there is a connected graph such that its phylogeny number minus its competition number is greater than k.We get similar results for hypergraphs.     

Non-Jumping Numbers for 4-Uniform Hypergraphs

Let r≥2 be an integer. A real number α ∈ [0,1) is a jump for r if for any Open image in new window >0 and any integer m, m≥r, any r-uniform graph with n>n₀( Open image in new window ,m) vertices and at least Open image in new window edges contains a subgraph with m vertices and at least Open image in new window edges, where c=c(α) does not depend on Open image in new window and m. It follows from a theorem of Erd?s, Stone and Simonovits that every α ∈ [0,1) is a jump for r=2. Erd?s asked whether the same is true for r≥3. Frankl and Rödl gave a negative answer by showing that Open image in new window is not a jump for r if r≥3 and l>2r. Following a similar approach, we give several sequences of non-jumping numbers generalizing the above result for r=4.     

Some Properties of Chromatic Coefficients of Linear Uniform Hypergraphs

In this paper a formula for the coefficient of λ ^n-gh+g-1 in the chromatic polynomial of a linear h-uniform hypergraph H of order n and girth g is proposed provided (g,  h) ≠ (3, 3). This formula involves 13 types of spanning subhypergraphs of H for 3 ≤ h ≤ 5.