Given a fixed $k$-uniform hypergraph $F$, the $F$-removal lemma states that every hypergraph with few copies of $F$ can be made $F$-free by the removal of few edges. Unfortunately, for general $F$, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which $F$ one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when $F$ is $k$-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon's result can be extended to all $k>2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when $F$ is $k$-partite. In this paper we prove this conjecture.
The classical sharp threshold theorem of Friedgut and Kalai (1996) asserts that any symmetric monotone function $f:\{0,1\}^{n}\to\{0,1\}$ exhibits a sharp threshold phenomenon. This means that the expectation of $f$ with respect to the biased measure $μ_{p}$ increases rapidly from 0 to 1 as $p$ increases. In this paper we present `robust' versions of the theorem, which assert that it holds also if the function is `almost' monotone, and admits a much weaker notion of symmetry. Unlike the original proof of the theorem which relies on hypercontractivity, our proof relies on a `regularity' lemma (…
The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.
We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ has no induced copies of $(W,H,\prec)$. (Note that we do \emph{not} need to modify the ordering $<$.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstr…
We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Král', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemerédi's theorem:…
In this paper, we define a homogeneous polynomial for a general hypergraph, and establish a remarkable connection between clique number and the homogeneous polynomial of a general hypergraph. For a general hypergraph, we explore some inequality relations among spectral radius, clique number and the homogeneous polynomial. We also give lower and upper bounds on the spectral radius in terms of the clique number.
In this paper, we prove that $(-\infty, 0)$ is a zero-free interval for chromatic polynomials of a family ${\cal L}_0$ of hypergraphs and $(0, 1)$ is a zero-free interval for chromatic polynomials of a subfamily ${\cal L}_0'$ of ${\cal L}_0$ of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.
In this paper, we give a reduced formula of the characteristic polynomial of $k$-uniform hypergraphs with a pendant edge. And the explicit characteristic polynomial and all distinct eigenvalues of $k$-uniform hyperpath are given.
In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for…
T. Kálmán (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and $(1,1/y)$ to hypergraphs. The two polynomials are defined under a fixed ordering of hyperedges, and are proved to be independent of the ordering using techniques of polytopes. In this paper, similar to the Tutte's original proof we provide a direct and elementary proof for the well-definedness of the interior and exterior polynomials of hypergraphs.
Let $G = (V,E)$ be a finite, simple, connected graph with chromatic polynomial $P_G(q)$. Sokal \cite{sokal} proved that the roots of the chromatic polynomial of $G$ are bounded in absolute value by $KD$ where, $D$ is the maximum degree of the graph and $7< K < 8$ is a constant. In this paper we generalize this result to uniform hypergraphs. To prove our results we will use the theory of the bounded exponential type graph polynomials.
Exact minimum degree thresholds for perfect matchings in uniform hypergraphs
给定正整数k和r，其中4整除k且k/2 \leq r \leq k-1，我们给出了k-一致超图中保证完美匹配的最小r-度条件.这个条件是最好的可能和改进的工作Pikhurko谁给了一个渐近精确的结果。我们的方法利用吸收的方法，以及超图删除引理和Keevash和Sudakov的结构结果有关的图兰数的扩展三角形。
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly.
We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollobás, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by their chromatic polynomials. We show that for $r$-uniform hypergraphs with $r \geq 3$ this is almost never the case. This disproves the analolgue of the BPR-conjecture for $3$-uniform hypergraphs. For $r =2$ this also holds for the independence polynomial, as shown by J.A. Makowsky and V. Rakita (2017), whereas for the chromatic and matching polynomial this remain…
In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
Given an integer $k\geq 3$ and an initial $k-1$ isolated vertices, an {\em antiregular $k$-hypergraph} is constructed by alternatively adding an isolated vertex (connected to no other vertices) or a dominating vertex (connected to every other $k-1$ vertices). Let $a_i$ be the number of independent sets of cardinality $i$ in a hypergraph $H$, then the {\em independence polynomial} of $H$ is defined as $I(H;x)=\sum_{i=0}^m a_i x^i$, where $m$ is the size of a maximum independent set. The main purpose of the present paper is to generalise some results of independence polynomials of antiregular gr…
We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.
We present a general framework to represent discrete configuration systems using hypergraphs. This representation allows one to transfer combinatorial removal lemmas to their analogues for configuration systems. These removal lemmas claim that a system without many configurations can be made configuration-free by removing a few of its constituent elements. As applications of this approach we give, an alternative proof of the removal lemma for permutations by Cooper, a general version of a removal lemma for linear systems in finite abelian groups, an interpretation of the mentioned removal lemm…
For any $r$-uniform hypergraph $\mathcal{H}$ with $m$ ($\geq 2$) edges, let $P(\mathcal{H},k)$ and $P_l(\mathcal{H},k)$ be the chromatic polynomial and the list-color function of $\mathcal{H}$ respectively, and let $ρ(\mathcal{H})$ denote the minimum value of $|e\setminus e'|$ among all pairs of distinct edges $e,e'$ in $\mathcal{H}$. We will show that if $r\ge3$, $ρ(\mathcal{H})\ge 2$ and $m\ge \frac{ρ(\mathcal{H})^3}2+1$, then $P_l(\mathcal{H},k)=P(\mathcal{H},k)$ holds for all integers $k\geq \frac{2.4(m-1)}{ρ(\mathcal{H})\log(m-1)}$.
Let $H$ be a $k$-graph on $n$ vertices, with minimum codegree at least $n/k + cn$ for some fixed $c > 0$. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in $H$ or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of $n/k - cn$. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.
Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi …
Let $D_2$ denote the $3$-uniform hypergraph with $4$ vertices and $2$ edges. Answering a question of Alon and Shapira, we prove an induced removal lemma for $D_2$ having polynomial bounds. We also prove an Erdős-Hajnal-type result: every induced $D_2$-free hypergraph on $n$ vertices contains a clique or an independent set of size $n^{c}$ for some absolute constant $c > 0$. In the case of both problems, $D_2$ is the only nontrivial $k$-uniform hypergraph with $k\geq 3$ which admits a polynomial bound.
We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.
On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group, Institute for Basic Science（基础科学研究院极值组合与概率组）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Korteweg-de Vries Institute for Mathematics, University of Amsterdam（科特韦格-德弗里斯数学研究所，阿姆斯特丹大学）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；Dutch Institute for Emergent Phenomena（荷兰涌现现象研究所）
A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa-Szemerédi triangle removal lemma, which states that if a graph is $\varepsilon$-far from being triangle free, then most subsets of vertices of size $C(\varepsilon)$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon)$ is given by a tower-type function, and it is known that $C(\varepsilon)$ is not polynomial in $\varepsilon^{-1}$. The triangle removal lemma has been extended to many ot…
In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
The $k$-power hypergraph $G^{(k)}$ is the $k$-uniform hypergraph that is obtained by adding $k-2$ new vertices to each edge of a graph $G$, for $k \geq 3$. A parity-closed walk in $G$ is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of $G^{(k)}$ using the eigenvalues of signed subgraphs of $G$. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of $G^{(k)}$ in terms of parity-closed walks of $G$. Moreover, we give an explicit expression for the …
We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random $k$-uniform hypergraph $H_{n,p;k}$, for a wide range of $p$. We also show that it is solvable w.h.p. for random $r$-regular, $k$-uniform hypergraphs $H_{n,r;k},r=O(1)$.
DP color functions versus chromatic polynomials for hypergraphs (I)
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）
