Uniform Hypergraphs Research Articles

Polynomial Removal Lemma for Ordered Matchings

We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.

The Algebraic Multiplicity of the Spectral Radius of a Uniform Hypertree

It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and the matching polynomials to give the algebraic multiplicity of the spectral radius of a uniform hypertree.

Turán densities for daisies and hypercubes

AbstractAn ‐daisy is an ‐uniform hypergraph consisting of the six ‐sets formed by taking the union of an ‐set with each of the 2‐sets of a disjoint 4‐set. Bollobás, Leader and Malvenuto, and also Bukh, conjectured that the Turán density of the ‐daisy tends to zero as . In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Turán densities of hypercubes. For fixed and large , we show that the smallest set of vertices of the ‐dimensional hypercube that intersects every copy of has asymptotic density strictly below , for all . In fact, we show that this asymptotic density is at most , for some constant . As a consequence, we obtain similar bounds for the edge‐Turán densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities.

Tight 9-Cycle Decompositions of λ-Fold Complete 3-Uniform Hypergraphs

For 2≤t≤m, let Zm denote the group of integers modulo m, and let TCm(t) denote the t-uniform hypergraph with vertex set Zm and hyperedge set {{i,i+1,i+2,…,i+t−1}:i∈Zm}. Any hypergraph isomorphic to TCm(t) is a t-uniform tight m-cycle. In this paper, we consider the existence of tight 9-cycle decompositions of λ-fold complete 3-uniform hypergraphs. According to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, we obtain the necessary and sufficient conditions for the existence of TC9(3)-decomposition of λKn(3). We show there exists a TC9(3)-decomposition of λKn(3) if and only if λn(n−1)(n−2)≡0(mod54), λ(n−1)(n−2)≡0(mod6) and n≥9.

On the Ramsey numbers of daisies I

Abstract Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An $r$ -daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$ -uniform hypergraph $\{K\cup P\,{:}\, P\in M^{(r)}\}$ . Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$ -daisies and also for special cases where the size of the kernel is bounded.

On the Ramsey numbers of daisies II

Abstract A $(k+r)$ -uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$ -daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$ , $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$ -tuples $K\cup P$ , where $P$ is an $r$ -tuple of $M$ . We obtain an $(r-2)$ -iterated exponential lower bound to the Ramsey number of an $(r,m,k)$ -daisy for $2$ -colours. This matches the order of magnitude of the best lower bounds for the Ramsey number of a complete $r$ -graph.

An algebraic approach to the reconstruction of uniform hypergraphs from their degree sequence

The reconstruction of a (hyper)graph starting from its degree sequence is one of the most relevant among the inverse problems investigated in the field of graph theory. In case of graphs, a feasible solution can be quickly reached, while in case of hypergraphs Deza et al. (2018) proved that the problem is NP-hard even in the simple case of 3-uniform ones. This result opened a new research line consisting in the detection of instances for which a solution can be computed in polynomial time. In this work we deal with 3-uniform hypergraphs, and we study them from a different perspective, exploiting a connection of these objects with partially ordered sets. More precisely, we introduce a simple partially ordered set, whose ideals are in bijection with a subclass of 3-uniform hypergraphs. We completely characterize their degree sequences in case of principal ideals (here a simple fast reconstruction strategy follows), and we furthermore carry on a complete analysis of those degree sequences related to the ideals with two generators. We also consider unique hypergraphs in Dext, i.e., those hypergraphs that do not share their degree sequence with other non-isomorphic ones. We show that uniqueness holds in case of hypergraphs associated to principal ideals, and we provide some examples of hypergraphs in Dext where this property is lost.

Heterogeneous dense subhypergraph detection

AbstractWe study the problem of testing the existence of a heterogeneous dense subhypergraph. The null hypothesis corresponds to a heterogeneous Erdös–Rényi uniform random hypergraph and the alternative hypothesis corresponds to a heterogeneous uniform random hypergraph that contains a dense subhypergraph. We establish detection boundaries when the edge probabilities are known and construct an asymptotically powerful test for distinguishing the hypotheses. We also construct an adaptive test which does not involve edge probabilities, and hence, is more practically useful.

On stabilizing index and cyclic index of certain amalgamated uniform hypergraphs

Let G be a connected uniform hypergraph and A(G) be the adjacency tensor of G. The largest absolute value of the eigenvalues of A(G) is called the spectral radius of G. The number of eigenvectors of A(G) associated with the spectral radius is called the stabilizing index of G. The number of eigenvalues of A(G) with modulus equal to the spectral radius is called the cyclic index of G. In this paper, we consider a class of amalgamated uniform hypergraphs and compute its stabilizing index and cyclic index.

Regular Subgraphs of Linear Hypergraphs

Abstract We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Łuczak, Mubayi, Nagle, Person, Rödl, Schacht, and Verstraëte. We use this result to show that the maximum number of edges in a $3$-uniform hypergraph on $n$ vertices containing no immersion of a closed surface is $n^{2+o(1)}$. Furthermore, we present results on the maximum number of edges in $k$-uniform linear hypergraphs containing no $r$-regular subhypergraph.

Constructing cospectral hypergraphs

Spectral hypergraph theory mainly concerns using hypergraph spectra to obtain structural information about the given hypergraphs. The study of cospectral hypergraphs is important since it reveals which hypergraph properties cannot be deduced from their spectra. In this paper, we show a new method for constructing cospectral uniform hypergraphs using two well-known hypergraph representations: adjacency tensors and adjacency matrices.

On harmonious coloring of hypergraphs

A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.

The growth rate of multicolor Ramsey numbers of 3-graphs

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

A note on the existence of regular r -partite self-complementary and almost self-complementary r -uniform hypergraphs

A note on the existence of regular r -partite self-complementary and almost self-complementary r -uniform hypergraphs

An Overview on Self-Complementary 3-Uniform Hypergraphs

An Overview on Self-Complementary 3-Uniform Hypergraphs

Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles

AbstractIn this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let denote the maximum number of edge‐disjoint copies of a fixed simple graph that can be placed on an ‐vertex ground set without forming a subgraph whose edges are from different ‐copies. The case when both and are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when and are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl, and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear ‐uniform hypergraph Turán problems to determine form a class of the multicolor Turán problem, following the identity , our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every up to a subpolynomial factor. Furthermore, when is a triangle, we settle the case and give bounds for the cases , as well.

On the number of A-transversals in hypergraphs

A set S of vertices in a hypergraph is strongly independent if every hyperedge shares at most one vertex with S. We prove a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph analogous to the Moon-Moser theorem. Given an r-uniform hypergraph H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}$$\\end{document} and a non-empty set A of non-negative integers, we say that a set S is an A-transversal of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}$$\\end{document} if for any hyperedge H of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}$$\\end{document}, we have |H∩S|∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|H\\cap S| \\in A$$\\end{document}. Independent sets are {0,1,⋯,r-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1,\\dots ,r{-}1\\}$$\\end{document}-transversals, while strongly independent sets are {0,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}$$\\end{document}-transversals. Note that for some sets A, there may exist hypergraphs without any A-transversals. We study the maximum number of A-transversals for every A, but we focus on the more natural sets, A={a}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A=\\{a\\}$$\\end{document}, A={0,1,⋯,a}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A=\\{0,1,\\dots ,a\\}$$\\end{document} or A being the set of odd integers or the set of even integers.

Incorporating reputation into reinforcement learning can promote cooperation on hypergraphs

Most previous studies that investigate the evolutionary dynamics of cooperation in the multi-agent social dilemma only consider the pairwise interactions. However, pairwise interactions alone cannot adequately characterize the higher-order interaction relationships that are commonly present in real-world social systems. In this study, we propose a spatial public goods game on uniform random hypergraphs with reputation mechanism and self-regarding Q-learning algorithm, where agents take an action based on the Q-value and update their reputation by the first-order evaluation norm. We define the reward as a function of the agent’s payoff and their reputation influenced by the social attention degree. Simulation results indicate that an increase in the social attention degree can effectively facilitate cooperation, as social reputation incentivizes agents to adopt cooperative actions to obtain optimal rewards. Additionally, we find that the discount factor and the learning rate both significantly impact the evolution of cooperation, with the influence of the learning rate on the evolution of cooperation being dependent on the discount factor. Finally, the robustness of this model is demonstrated based on the results of theoretical analysis.

On ABC spectral radius of uniform hypergraphs

On ABC spectral radius of uniform hypergraphs

Construction of S(3)(2,3)-Designs of Any Index

Let H(3) be a uniform hypergraph of rank 3. A hyperstar S(3)(2,3) of centreC={x,y} is a 3-uniform hypergraph with three hyperedges, all having the centreC={x,y} in common, with x and y of degree 3 and the remaining vertices of degree 1. In this paper, we determine the spectrum of S(3)(2,3)-designs for any index λ.