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.     

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.     

完全3-一致超图K_n~((3))的哈密顿圈分解

基于王建方和李东给出的超图哈密顿圈的定义和Katona-Kierstead给出的超图哈密顿链的定义,近年来,国内外学者对一致超图的哈密顿圈分解的研究有一系列结果.特别是Bailey-Stevens和Meszka-Rosa研究了完全3-一致超图K_n~((3))的哈密顿圈分解,得到了n=6k+1,6k+2(k=1,2,3,4,5)的哈密顿圈分解.本文在吉日木图提出的边划分方法的基础上继续研究,得到了完全3-一致超图K_n~((3))的哈密顿圈分解的算法,由此得到了n=6k+2,6k+4(k=1,2,3,4,5,6,7),n=6k+5(k=1,2,3,4,5,6)时的圈分解.这一结果将Meszka-Rosa关于K_n~((3))的哈密顿圈分解结果从n≤32提高到了n≤46(n≠43).     

Homology Cycles and Dependent Cycles of Hypergraphs

In this paper, some new concepts for hypergraphs are introduced. Based on the previous results, we do further research on cycle structures of hypergraphs and construct a more strictly complete cycle structure system of hypergraphs.     

Monochromatic Matchings in the Shadow Graph of Almost Complete Hypergraphs

Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e., on the pairs that are covered by hyperedges. We show that in any (r – 1)-coloring of the edges of an r-uniform hypergraph with n vertices and at least (1-e)( *20c nr)(1-\varepsilon)\left( {\begin{array}{*{20}c} n\\ r\\ \end{array}}\right) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result confirms an earlier conjecture and implies that for any fixed r and sufficiently large n, there is a monochromatic Berge-cycle of length (1 – o(1))n in every (r – 1)-coloring of the edges of K^(r)_n{K^{(r)}_{n}}, the complete r-uniform hypergraph on n vertices.     

Partitioning 2‐Edge‐Colored Ore‐Type Graphs by Monochromatic Cycles

Consider a graph G on n vertices satisfying the following Ore‐type condition: for any two nonadjacent vertices x and y of G, we have . We conjecture that if we color the edges of G with two colors then the vertex set of G can be partitioned to two vertex disjoint monochromatic cycles of distinct colors. In this article, we prove an asymptotic version of this conjecture.     

On Extremal Hypergraphs for Hamiltonian Cycles

We study sufficient conditions for Hamiltonian cycles in hypergraphs and obtain both Turán- and Dirac-type results. While the Turán-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields just a sufficient condition relying solely on the minimum vertex degree.     

Coloring Non-uniform Hypergraphs Without Short Cycles

The work deals with a generalization of Erd?s–Lovász problem concerning colorings of non-uniform hypergraphs. Let H  = (V, E) be a hypergraph and let \({{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}\) for some r ≥ 2. Erd?s and Lovász proposed to find the value f (n) equal to the minimum possible value of f ₂(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n ≥ 3 and girth at least 4, satisfying the inequality $$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$ then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.     

Approximations to m‐Colored Complete Infinite Hypergraphs

Given an edge coloring of a graph with a set of m colors, we say that the graph is exactly m‐colored if each of the colors is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number and for all sufficiently large k, there is a k‐coloring of the complete graph on such that no complete infinite subgraph is exactly m‐colored. In the light of this result, we consider the question of how close we can come to finding an exactly m‐colored complete infinite subgraph. We show that for a natural number m and any finite coloring of the edges of the complete graph on with m or more colors, there is an exactly ‐colored complete infinite subgraph for some satisfying ; this is best possible up to the additive constant. We also obtain analogous results for this problem in the setting of r‐uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalizations of this conjecture to r‐uniform hypergraphs.     

完全图中的正常染色的路和圈

令$K_{n}^{c}$表示$n$ 个顶点的边染色完全图.
令 $\Delta^{mon}
(K_{n}^{c})$表示$K^c_{n}$的顶点上关联的同种颜色的边的最大数目.
如果$K_{n}^{c}$中的一个圈（路）上相邻的边染不同颜色,则称它为正常染色的.
B. Bollob\'{a}s和P. Erd\"{o}s (1976) 提出了如下猜想：若 $\Delta^{{mon}}
(K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, 则$K_{n}^{c}$中含有一个正常染
色的Hamilton圈. 这个猜想至今还未被证明.我们研究了上述条件下的正常染色的路和圈.     

Existence of Monochromatic Simplices of Given Volume in a Colored Rational Lattice

Mathematical Notes -     

On Edge-Colored Graphs Covered by Properly Colored Cycles

We characterize edge-colored graphs in which every edge belongs to some properly colored cycle. We obtain our result by applying a characterization of 1-extendable graphs. Received: April, 2003     

The Monochromatic Circumference of 2‐Edge‐Colored Graphs

Li et al. (Discrete Math 310 (2010), 3579–3583) asked how long the longest monochromatic cycle in a 2‐edge‐colored graph G with minimum degree at least could be. In this article, an answer is given for all to an asymptotic form of their question.     

Decomposing Berge Graphs Containing No Proper Wheel, Long Prism Or Their Complements

In this paper we show that, if G is a Berge graph such that neither G nor its complement contains certain induced subgraphs, named proper wheels and long prisms, then either G is a basic perfect graph( a bipartite graph, a line graph of a bipartite graph or the complement of such graphs) or it has a skew partition that cannot occur in a minimally imperfect graph. This structural result implies that G is perfect. This work was supported in part by NSF grant DMI-0352885 and ONR grant N00014-03-1-0188.     

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 .     

4一致反超图的最小边数问题

混合超图是在超图的基础上添加一个反超边得到的图．超边和反超边的区别主要体现在着色要求上．在着色中,要求每一超边至少要有两个点着不同的颜色,而每一反超边至少有两个点着相同的颜色．最大最小颜色数分别称为混合超图的上色数和下色数。本文主要研究反超图,即只含反超边的超图。讨论了上色数为3的4一致超图的最小边数问题．给出了上色数为3的4一致反超图的最小边数的一个上界和一个下界．