Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Coloring uniform hypergraphs with few edges

A hypergraph is b‐simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r‐uniform non‐t‐colorable hypergraph of girth at least g. Erd?s and Lovász proved that A result of Szabó improves the lower bound by a factor of r^2?? for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b‐simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and ? > 0 and sufficiently large r, every r‐uniform b‐simple hypergraph with maximum edge‐degree at most t^rr^1?? is t‐colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Tight Hamilton cycles in random uniform hypergraphs

In this paper we show that e/n is the sharp threshold for the existence of tight Hamilton cycles in random k ‐uniform hypergraphs, for all k ≥ 4. When k = 3 we show that 1/n is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

D-完全一致混合超图不可着色的一个充要条件

混合超图的上,下色数与C-超边和D-超边数有着必然联系.一般地,增加C边会使下色数x(H)增加,增加D-超边会使上色数(x)(H)减小.本论文对D-完全一致混合超图进行研究,利用组合数学中分划思想及方法得到的D-完全一致混合超图不可着色的一个充要条件,对D-完全一致混合超图能否着色找到了可行的依据,进一步揭示C-超边数...     

Coloring BIBDs with block size 4

It is shown that for each integer m ≥ 1 there exists a lower bound, v_m, with the property that for all v ≥ v_m with v ≡ 1, 4 (mod 12) there exists an m-chromatic S(2, 4, v) design. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 403–409, 1998     

On Ramsey numbers of uniform hypergraphs with given maximum degree

For every ?>0 and every positive integers Δ and r, there exists C=C(?,Δ,r) such that the Ramsey number, R(H,H) of any r-uniform hypergraph H with maximum degree at most Δ is at most C|V(H)|^1+?.     

Greedy colorings of uniform hypergraphs

We give a very short proof of an Erd?s conjecture that the number of edges in a non‐2‐colorable n‐uniform hypergraph is at least f(n)2ⁿ, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) ≥ cn^1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) ≥ cn^1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k‐colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n‐uniform, n‐regular hypergraphs are 2‐colorable if n ≥ 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Isomorphic edge disjoint subgraphs of hypergraphs

We show that any k‐uniform hypergraph with n edges contains two isomorphic edge disjoint subgraphs of size for k = 4, 5 and 6. This is best possible up to a logarithmic factor due to an upper bound construction of Erd?s, Pach, and Pyber who show there exist k‐uniform hypergraphs with n edges and with no two edge disjoint isomorphic subgraphs with size larger than . Furthermore, our result extends results Erd?s, Pach and Pyber who also established the lower bound for k = 2 (eg. for graphs), and of Gould and Rödl who established the result for k = 3. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 767–793, 2016     

Partitions of hypergraphs under variable degeneracy constraints

The paper deals with partitions of hypergraphs into induced subhypergraphs satisfying constraints on their degeneracy. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph $H$ and a sequence $f=\left(f_1,f_2,\text{…},f_p\right)$ of $p\ge 1$ vertex functions $f_i:V\left(H\right)\to {\mathbb{N}}_0$ such that $f_1\left(v\right)+f_2\left(v\right)+\mathrm{?}+f_p\left(v\right)\ge d_H\left(v\right)$ for all $v\in V\left(H\right)$, we want to find a sequence $\left(H_1,H_2,\text{…},H_p\right)$ of vertex disjoint induced subhypergraphs containing all vertices of $H$ such that each hypergraph $H_i$ is strictly $f_i$‐degenerate, that is, for every nonempty subhypergraph $H\prime ?H_i$ there is a vertex $v\in V\left(H\prime \right)$ such that $d_{H\prime }\left(v\right)<f_i\left(v\right)$. Our main result in this paper says that such a sequence of hypergraphs exists if and only if $\left(H,f\right)$ is not a so‐called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. Our main result has several interesting applications related to generalized hypergraph coloring problems.     

Packing tight Hamilton cycles in 3‐uniform hypergraphs

Let H be a 3‐uniform hypergraph with n vertices. A tight Hamilton cycle C ? H is a collection of n edges for which there is an ordering of the vertices v₁,…,v_n such that every triple of consecutive vertices {v_i,v_i+1,v_i+2} is an edge of C (indices are considered modulo n ). We develop new techniques which enable us to prove that under certain natural pseudo‐random conditions, almost all edges of H can be covered by edge‐disjoint tight Hamilton cycles, for n divisible by 4. Consequently, we derive the corollary that random 3‐uniform hypergraphs can be almost completely packed with tight Hamilton cycles whp, for n divisible by 4 and p not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo‐random digraphs with even numbers of vertices. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Monochromatic loose path partitions in k-uniform hypergraphs

A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k?2$ vertices. A weaker form of this conjecture with $2k?5$ uncovered vertices instead of $k?2$ is proved. Thus the conjecture holds for $k=3$. The main result of this paper states that the conjecture is true for all $k\ge 3$.     

Loose Hamilton cycles in hypergraphs

We prove that any k-uniform hypergraph on n vertices with minimum degree at least contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.     

极大限制边连通超图的两个充分条件

图的限制边连通度是经典边连通度的推广,可用于精确度量网络的容错性.极大限制边连通图是使限制边连通度达到最优的一类图.首先将图的限制边连通度和最小边度的概念推广到r一致线性超图H,证明当H的最小度δ(H)≥r+1时,H的最小边度ξ(H)是它的限制边连通度λ′(H)的一个上界,并将满足ξ(H)=λ′(H)的H称为极大限制边连通超图,然后证明n个顶点的r一致线性超图H如果满足δ(H)≥(n-1)/(2(r-1))+(r-1),则它是极大限制边连通的,最后证明直径为2,围长至少为4的一致线性超图是极大限制边连通的.所得结论是图中相关结果的推广.     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008