正则多部超图的计数

Extremal problems for ordered hypergraphs: small patterns and some enumeration

We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with at most four incidences (we give proofs only for six cases). We prove some exact formulae and recurrences for the numbers of hypergraphs, simple and all, with n incidences and derive rough logarithmic as…

Asymptotic enumeration of sparse uniform linear hypergraphs with given degrees

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main resu…

On the König-Hall-Egerváry theorem for multidimensional matrices and multipartite hypergraphs

One of possible interpretations of the well-known König--Hall--Egerváry theorem is a full characterization of all bipartite graphs extremal for fractional matchings of a given weight (or, equivalently, a characterization of $(0,1)$-matrices extremal for partial fractional diagonals of a given length). In this paper we initiate the study of $d$-partite $d$-uniform hypergraphs that are extremal for fractional perfect matchings (or, equivalently, $d$-dimensional $(0,1)$-matrices that are extremal for polydiagonals). For this purpose, we analyze similarities and differences between $2$-dimensional…

Asymptotic enumeration of linear hypergraphs with given number of vertices and edges

For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ \frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \frac32})$. We also find the probability that a random $r$-uniform lin…

On the number of linear multipartite hypergraphs with given size

For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with $r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. Let $A_1,\ldots,A_k$ be a given $k$-partition of $[n]$ with $|A_i|=n_i\geqslant 1$. An $r$-uniform hypergraph $H$ is called {\it $k$-partite} if each edge $e$ satisfies $|e\cap A_i|\leqslant 1$ for $1\leqslant i\leqslant k$. In this paper, the number of linear $k$-partite $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the numbe…

Bounded colorings of multipartite graphs and hypergraphs

Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollobás and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruciński. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph $G$. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal suf…

Monochromatic Edges in Complete Multipartite Hypergraphs

Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boy, all girl and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when ge…

A multipartite version of the Hajnal-Szemerédi theorem for graphs and hypergraphs

A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemerédi states that if $G$ is a graph of order $n$ with minimum degree $δ(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + γ)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore…

The edge ideals of complete multipartite hypergraphs

We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, $l$-Cohen-Macaulay, $l$-Buchsbaum, unmixed, and satisfying Serre's condition $S_r$, via some combinatorial terms. Also, we prove that for a complete $s$-uniform $t$-partite hypergraph $H$, vertex decomposability, shellability, sequentially $S_r$ and sequentially Cohen-Macaulay properties coincide with the condition that $H$ has $t-1$ sides consisting of a single vertex. Moreover, we show that the latter condition occurs if and onl…

多重部超图的可选择性

Choosability of multipartite hypergraphs

本文研究了多重部超图的可选择性，证明了在特定条件下，此类超图可以被高效着色。

List colourings of multipartite hypergraphs

Let $χ_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then $χ_l(G)\ge (1/(r-1)+o(1))\log_r d$. To see how close this inequality is to best possible, we examine $χ_l(G)$ when $G$ is a random $r$-partite hypergraph with $n$ vertices in each class. The value when $r=2$ was determined by Alon and Krivelevich, here we show that $χ_l(G)= (g(r,α)+o(1))\log_r d$ almost surely, where $d$ is the expected average degree of~$G$ and $α…

On Ryser's Conjecture for Linear Intersecting Multipartite Hypergraphs

Ryser conjectured that $τ\le(r-1)ν$ for $r$-partite hypergraphs, where $τ$ is the covering number and $ν$ is the matching number. We prove this conjecture for $r\le9$ in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each $r$-partite hypergraph should have a cover of size $(r-1)ν$ of a particular form. We provide a counterexample to Aharoni's conjecture with $r=13$ and $ν=1$. We also report a number of computational results. For $r=7$, we …

Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}…

多部图与超图的奇 Ramsey 数

Odd Ramsey numbers of multipartite graphs and hypergraphs

本文研究多部图与超图的奇 Ramsey 数，推广已有结果并确定两类结构的渐近值，其中超图情形属首次。

计数具有指定度数和边类型的二超图

Enumeration of dihypergraphs with specified degrees and edge types

本文研究了具有给定入度序列、出度序列及超边头尾大小的定向超图的计数问题，给出了渐近公式。

Tiling multipartite hypergraphs in Quasi-random Hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $Ω(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $Ω(n^2)$ and minimum codegree $Ω(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense …

多部超图的不变覆盖

Invariant covers of multipartite hypergraphs

该研究证明了多部超图的不变覆盖定理，并扩展了Lovász估计的对称类比。

多部超图中的匹配

Matchings in multipartite hypergraphs

本文针对大型 $k$-部 $k$-一致超图确立了与二部图经典匹配定理类似的结论，回答了 Han、Zang 和 Zhao 的问题，并通过结合度与重数条件研究彩虹匹配给出了证明。

Factorizations of Complete Multipartite Hypergraphs

In a mathematics workshop with $mn$ mathematicians from $n$ different areas, each area consisting of $m$ mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly $r$ meeting(s) each day, and (iii) each pair of people of the same area have exactly $λ$ meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In partic…

Asymptotic enumeration of sparse connected 3-uniform hypergraphs

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=ω(N^{1/3}\ln^{2} N)$. This almost completely fills the gap in the range of $M$ for which the formula is known. We approach the problem using an `inside-out' approach of an earlier paper of Pittel and the second author, for connected graphs. A key part of the method uses structural components of connected hypergraphs called cores and kernels. These are structural components of connected hypergraphs. Our results also give i…

多部超图的弱饱和性

Weak saturation of multipartite hypergraphs

该研究在有向情形下精确确定完全多部q-图的弱饱和数，推广Alon的经典定理，同时建立二部图在不同宿主图中弱饱和数的联系并渐近确定团中完全q部q-图的弱饱和数。

超图 Horn 函数中的余原子枚举：Horn 模型偏序集的秩-3 表示

Coatom Enumeration in Hypergraph Horn Functions: Rank-Three Representations of Horn Model Posets

该研究提出超图 Horn 函数中 Horn 模型偏序集的秩-3 表示，结合相关构造证明余原子枚举的复杂性，给出不同秩与频率下的生成算法及复杂度结论。

超图的有限VC维的极小覆盖集的枚举

Enumeration of minimal transversals of hypergraphs of bounded VC-dimension

本文证明了当超图的VC维有界时，极小覆盖集的枚举问题可以在多项式时间内解决，并扩展了已知的多项式情况。

超图卡塔兰数的公式与渐进行为

Formulas and asymptotics of hypergraph Catalan numbers

本文研究了k-游走的渐进行为，通过另一种方法验证了Gunnells关于超图卡塔兰数的渐近公式，发现其增长由星形树上的k-游走数量决定。

Degree sequences of sufficiently dense random uniform hypergraphs

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edg…

分形超树：组合枚举、对称性质与超度量结构

Fractal Hyper-Trees: Combinatorial Enumeration, Symmetry Properties, and Ultrametric Structures

本文提出并形式化分形超树（FHT），建立公理化基础，研究其组合枚举（关联OEIS A398931）、对称性质（自同构群）及超度量拓扑，并探讨渐近视角与$p$-进整数环的类比。

通过图容器和熵进行渐近计数

Asymptotic enumeration via graph containers and entropy

本文综述了结合图容器方法和熵方法在图同态计数中的最新发展。

On the number of linear hypergraphs of large girth

An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i, j\},\ i\neq j$. For every $r\geq 3$ and $\ell\geq 4$, we show that there exists a constant $C$ depending on $r$ and $\ell$ such that the number of linear $r$-graphs of girth $\ell$ is at most $2^{Cn^{1+1/\lfloor \ell/2\rfloor}}$. Furthermore, we extend the result for $\ell=4$, proving that there exists a constant $C$ depending on …

Supersaturated sparse graphs and hypergraphs

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are $2^{(1+o(1))\text{ex}(n,H)}$ such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs $H$. We make a first attempt at addressing this enumeration problem for a general bipartite graph $H$. We show that an upper bound of $2^{O…

Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsym…

Smirnov词与完全多部图中哈密顿圈的双射

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

建立平衡字母重数的Smirnov词与完全m部图K_{n,n,...,n}中哈密顿路径的双射，导出哈密顿圈数的闭式容斥公式，并推广到非均匀情形K_{n1,n2,...,nm}，给出渐近分析。

Erdős 问题 550 关于树与完全多部拉姆齐数的解决

A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers

本文解决了 Erdős 问题 550，证明了对固定整数 k≥2 和 1≤m1≤...≤mk，对任意足够大的 n 和任意 n 顶点树 T，有 R(T,K_{m1,...,mk}) ≤ (k-1)(R(T,K_{m1,m2})-1)+m1。

将拟阵编码到量子态中

Encoding matroids into quantum states

受图态和超图态公理框架启发，引入拟阵态公理构造，给出两种构造方法，建立通用全局算子，还展示如何由拟阵态得到图态，为描述多体量子态提供新语言。

Generalizations and strengthenings of Ryser's conjecture

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $ν(H)$, the vertex cover number is at most $(r-1)ν(H)$. This far reaching generalization of König's theorem is only known to be true for $r\leq 3$, or $ν(G)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $α(G)$, there exists at most $(r-1)α(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stron…

On the dualization in distributive lattices and related problems

In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal ``ideal solutions'' are to be generated. This in particular allows us to study the complexity of the problem un…

关于球体Turán数的条件下限

A conditional lower bound for the Turán number of spheres

该研究提出了关于球体Turán数的条件下界，并通过归纳法给出了上界，推测上界可与下界匹配。