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, we show that for any $k$, the set of independence densities of hypergraphs with hyperedges of size at most $k$ is closed and contains no infinite increasing sequences.
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…
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.
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.
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.
An old result of Müller and Rödl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and Rödl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm …
In this paper we study variations of an old result by Müller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers we can always find an infinite increasing path. We study corresponding questions for hypergraphs and directed graphs. For example we show that the condition that a hypergraph contains a subhypergraph with infinite degrees is equivalent to the condition that any vertex labeling contains an infinite increasing loose path. We also find an equivale…
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 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…
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 $…
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…
For a hypergraph $H=(V,\mathcal E)$, a subfamily $\mathcal C\subseteq \mathcal E$ is called a cover of the hypergraph if $\bigcup\mathcal C=\bigcup\mathcal E$. A cover $\mathcal C$ is called minimal if each cover $\mathcal D\subseteq\mathcal C$ of the hypergraph $H$ coincides with $\mathcal C$. We prove that for a hypergraph $H$ the following conditions are equivalent: (i) each countable subhypergraph of $H$ has a minimal cover; (ii) each non-empty subhypergraph of $H$ has a maximal edge; (iii) $H$ contains no isomorphic copy of the hypergraph $(ω,ω)$. This characterization implies that a coun…
The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random subgraphs of ${\rm KG}^2_{n,k}$ was recently addressed in a series of works by Bollob{á}s, Narayanan, and Raigorodskii (20…
A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous $k$-uniform hypergraphs which are not homogeneous (two with $k=3$, one with $k=4$, and one with $k=6$). Evidence is also given that these may be the only ones, up to complementation. For example, for $k=3$ …
Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $δ_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Turán density $γ(F)$ is the smallest $γ>0$ such that every $r$-graph on $n$ vertices with $δ_{r-1}(H)\ge (γ+ o(1))n$ contains $F$ as a subhypergraph. Using results on the independence number of hypergraphs, we show that there are constants $c_1, c_2>0$ depending only on $r$ such that \[ 1 - c_2 \frac{\ln t}{t^{r-1}} \le γ(K_t^r) \le 1 - c_1 \frac{\ln t}{t^{r-1}}, \] where $K_t^…
Let the {\it bicoloring cover number $χ^c(G)$} for a hypergraph $G(V,E)$ be the minimum number of bicolorings of vertices of $G$ such that every hyperedge $e\in E$ of $G$ is properly bicolored in at least one of the $χ^c(G)$ bicolorings. We investigate the relationship between $χ^c(G)$, matchings, hitting sets, $α(G)$(independence number) and $χ(G)$ (chromatic number). We design a factor $O(\frac{\log n}{\log \log n-\log \log \log n})$ approximation algorithm for computing a bicoloring cover. We define a new parameter for hypergraphs - "cover independence number $γ(G)$" and prove that $\log \f…
In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least $(1/p+o(1))n$, where $p$ is the smallest prime factor of $k$, then $H$ contains a perfect matching. Second, we show that if $H$ is a $3$-graph on $n\in 3\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ (the unique $3$-graph with $4$ vertices and $3$ edges) and has minimum codegree at least $(1/3+o(1)))n$, then $H$ contains a perfect matching. Mo…
Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices span at most $t$ edges. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os offered \$500 for a proof of this. For $k=3$ this has been settled for many values of $s$ including powers of three but it was not known for any $k\geq 4$ and $s\geq k+2$. Here we settle the conjecture for all $s \g…
In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathb…
