We obtain new lower bounds for the independence number of $K_r$-free graphs and linear $k$-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we also give a new short proof of the main result of \cite{CT91} using this approach. As byproducts, we also obtain some non-trivial identities involving binomial coefficients.
The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Prałat about independence densities of graphs. Furthermore, w…
We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence poly…
The independence polynomial of a hypergraph is the generating function for its independent (vertex) sets with respect to their cardinality. This article aims to discuss several recurrence relations for the independence polynomial using some vertex and edge operations. Further, an extension of the well-known recurrence relation for simple graphs to hypergraphs is proven and other novel recurrence relations are also discussed.
Let K_4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d) tending to infinity such that every 3-uniform, K_4-free hypergraph N vertices and average degree d has independence number at least ω(d) N/d^{1/2}. We answer this question by constructing a 3-uniform, K_4-free hypergraph with independence number at most 2N/d^{1/2}. We also provide counterexamples to several related conjectures and improve the lower bound of some hypergraph Ramsey numbers.
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…
Let $σ$ be a partition of the positive integer $r$. A $σ$-hypergraph $H=H(n,r,q|σ)$ is an $r$-uniform hypergraph on $nq$ vertices which are partitioned into $n$ classes $V_1, V_2, \ldots, V_n$ each containing $q$ vertices. An $r$-subset $K$ of vertices is an edge of the hypergraph if the partition of $r$ formed by the non-zero cardinalities $|K\cap V_i|, 1\leq i \leq n,$ is $σ$. In earlier works we have considered colourings of the vertices of $H$ which are constrained such that any edge has at least $α$ and at most $β$ vertices of the same colour, and we have shown that interesting results ca…
The independence number $α(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$α(H)\geq \sum_{u\in V(H)}f_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\geq 2$, where \begin{eqna…
The bounded degree complex $BD(G,λ)$ is a generalization of the matching complexes of a graph. In this paper, we show that the bounded degree complex of a forest is shellable, by using independence complexes of hypergraphs. We obtain a wedge decomposition result of bounded degree complexes when a graph $G$ has a leaf
Let $H=(V,E)$ be an $s$-uniform hypergraph of order $n$ and $k\geq 0$ be an integer. A $k$-independent set $S\subseteq H$ is a set of vertices such that the maximum degree in the hypergraph induced by $S$ is at most $k$. Denoted by $α_k(H)$ the maximum cardinality of the $k$-independent set of $H$. In this paper, we first give a lower bound of $α_k(H)$ by the maximum degree of $H$. Furthermore, we prove that $α_k(H)\geq \frac{s(k+1)n}{2d+s(k+1)}$ where $d$ is average degree of $H$, and $k\geq 0$ is an integer.
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the (lower-)associated independence hypergraph of the random hypergraph in terms of the probability distributions of certain random simplicial complex and certain random independence hypergraph of Erdös-Rényi type. We construct a graded structure of the map algebra explicitly and give algorithms to generate random hypergraphs and random simplicial complexes.
The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove new sharp bounds on the independence number of n-vertex (r+1)-uniform hypergraphs in which every r-element set is contained in at most d edges, where 0 < d < n/(log n)^{3r^2}. Our relatively short proof extends a method due to Shearer. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.
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…
We present a variant of a universality result of Rödl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].
We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].
Caro, West, and Yuster studied how $r$-uniform hypergraphs can be oriented in such a way that (generalizations of) indegree and outdegree are as close to each other as can be hoped. They conjectured an existence result of such orientations for sparse hypergraphs, of which we present a proof.
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.
The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring numb…
Using the Filmus-Golubev-Lifshitz method to bound the independence number of a hypergraph, we solve some problems concerning multiply intersecting families with biased measure. Among other results we obtain a stability result of a measure version of the Erdos-Ko-Rado theorem for multiply intersecting families.
Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.
We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
We prove the following: Fix an integer $k\geq 2$, and let $T$ be a real number with $T\geq 1.5$. Let $\cH=(V,\cE_2\cup \cE_3\cup\dots\cup\cE_k)$ be a non-uniform hypergraph with the vertex set $V$ and the set $\cE_i$ of edges of size $i=2,\ldots , k$. Suppose that $\cH$ has no $2$-cycles (regardless of sizes of edges), and neither contains $3$-cycles nor $4$-cycles consisting of $2$-element edges. If the average degrees $t_i^{i-1} := i |\cE_i|/ |V|$ satisfy that $t_i^{i-1} \leq T^{i-1} (\ln T)^{\frac{k-i}{k-1}}$ for $i= 2, \dots , k$, then there exists a constant $C_k > 0$, depending only on $…
An $r$-uniform hypergraph $H$ consists of a set of vertices $V$ and a set of edges whose elements are $r$-subsets of $V$. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph $H$ if it is a subhypergraph of $H$ which contains all vertices of $H$. Greenhill, Isaev, Kwan and McKay (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for $r$-uniform hypergraphs with given degree sequence $\boldsymbol{k} = (k_1,\ldots, k_n)$. Our formula holds when…
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
Let $G$ be an $r$-uniform hypergraph. When is it possible to orient the edges of $G$ in such a way that every $p$-set of vertices has some $p$-degree equal to $0$? (The $p$-degrees generalise for sets of vertices what in-degree and out-degree are for single vertices in directed graphs.) Caro and Hansberg asked if the obvious Hall-type necessary condition is also sufficient. Our main aim is to show that this is true for $r$ large (for given $p$), but false in general. Our counterexample is based on a new technique in sparse Ramsey theory that may be of independent interest.
A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3 and 4. Our work is motivated by a problem of Lo and Zhao, who asked for $k\ge 4$, how large of an independent set a $k$-graph $H$ on $n$ vertices necessarily has when its maximum $(k-2)$-degree $Δ_{k-2}(H)\le dn$. (The corresponding…
We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Δ$ have a zero-free disk almo…
The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Łojasiewicz property, we prove that the sequence of iterates generated…
Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple $r$-uniform hypergraphs with degree sequence $k$. (Here "simple" means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree $k_{\mathrm{max}}$ satisfies $k_{\mathrm{max}}^3 = o(M)$.
Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $Δ$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $Δ^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is $O( (Δ/\log f)^{1/(k-1)})$. This extends results of Frieze and the second author and Bennett and Bohman. A similar result is proved for 3-uniform hypergraphs where every vertex lies in few triangles. This generalizes a result of Alon, Krivelevich, and Sudakov, who proved the result for graphs. Our main new technical contribution is a deviation inequality for positive ra…
In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125-148] can be adapted from random graphs to random $r$-uniform hypergaphs and provide sufficient conditions when a random $r$-uniform hypergraph $\mathcal{H}^{(r)}(n,p)$ contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs. Moreover, we study univer…
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 give a short proof that any k-uniform hypergraph H on n vertices with bounded degree Δhas Ramsey number at most c(Δ, k)n, for an appropriate constant c(Δ, k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and Rödl. Moreover, our method demonstrates that, for k \geq 4, c(Δ, k) \leq 2^{2^{\Ddots^{2^{c Δ}}}}, where the tower is of height k and the constant c …
We investigate the independence number of two graphs constructed from a polarity of $\mathrm{PG}(2,q)$. For the first graph under consideration, the Erdős-Rényi graph $ER_q$, we provide an improvement on the known lower bounds on its independence number. In the second part of the paper we consider the Erdős-Rényi hypergraph of triangles $\mathcal{H}_q$. We determine the exact magnitude of the independence number of $\mathcal{H}_q$, $q$ even. This solves a problem posed by Mubayi and Williford.
A hypergraph $G=(V,E)$ is $(k,\ell)$-sparse if no subset $V'\subset V$ spans more than $k|V'|-\ell$ hyperedges. We characterize $(k,\ell)$-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lov{á}sz, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behaviour in terms of the sparsity parameters $k$ and $\ell$. Our constructions e…
We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.
We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big…
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…
A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n < m < n^{O(1)} then s(n,m)= m^2/w(m), where w(m) is a function that goes to infinity and is smaller than any po…
Characterization of sparse monotone graph classes with bounded domination-to-2-independence ratio
CNRS, LaBRI, Université de Bordeaux（法国国家科学研究中心，LaBRI，波尔多大学）；Charles University, Prague（布拉格查理大学）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
We establish that in the large degree limit, the value of certain optimization problems on sparse random hypergraphs is determined by an appropriate Gaussian optimization problem. This approach was initiated in Dembo et. al.(2016) for extremal cuts of graphs. The usefulness of this technique is further illustrated by deriving the optimal value for Max $q$-cut on Erdős-Rényi and random regular graphs, Max XORSAT on Erdős-Rényi hypergraphs, and the min-bisection for the Stochastic Block Model.
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…
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further …
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
