Uniform Hypergraphs Research Articles

Some results and problems on clique coverings of hypergraphs

AbstractFor a ‐uniform hypergraph we consider the parameter , the minimum size of a clique cover of the edge set of . We derive bounds on for belonging to various classes of hypergraphs.

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

This paper continues the study of second-order signature properties—the characterization of the extreme 3-uniform hypergraph. Previously, bases were used to count extreme 3-uniform hypergraphs. However, the algorithm using this mechanism is extremely labor-intensive. The structure of the signature allows us to use it as a more efficient basis for the same problem. Here, we establish the nature of the mutual correspondence between the kind of second-order signature and extreme hypergraphs, and we present a new algorithm to find the power of the set of extreme 3-uniform hypergraphs through the set of their characteristic-signatures. New results obtained with the proposed tool are also presented.

Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence.

Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k - 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ2 testing of contingency tables. We have shown that this hypergraph-based χ2 test is more sensitive than the standard χ2 test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

Maximum Packing of λ-Fold Complete 3-Uniform Hypergraph with a Special Tetrahedron

Let Kv(3)=(V,E) be the complete 3-uniform hypergraph, where the vertex set is V={x1,x2,⋯,xv}, in which the edge set E is of all triples. Let ST denote the special tetrahedron with four edges, where each edge contains three vertices of degree 2. In this paper, we consider the decomposition and packing of a complete 3-uniform hypergraph of an λ-fold special tetrahedron. Firstly, the necessary conditions for the existence of the λ-fold ST-decomposition are discussed in four distinct cases. Secondly, 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, it is proven that the above necessary conditions are sufficient. Finally, we prove that a maximum ST-packing of a complete 3-uniform hypergraph Kv(3) exists for all v≥6 and λ≥1.

Generalizations of Cayley graphs to uniform hypergraphs

Generalizations of Cayley graphs to uniform hypergraphs

Distributed Optimization of Multiagent Systems Over Uniform Hypergraphs

Distributed Optimization of Multiagent Systems Over Uniform Hypergraphs

Conflict‐free hypergraph matchings

AbstractA celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection of subsets . We say that a matching is conflict‐free if does not contain an element of as a subset. Under natural assumptions on , we prove that has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called ‘high‐girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict‐free matching process’.

Extremal Numbers and Sidorenko’s Conjecture

Abstract Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\textrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

Wickets in 3-uniform hypergraphs

In this note, we consider a Turán-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let Hn(3) be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called wicket, is formed by three rows and two columns of a 3×3 point matrix. We describe two linear hypergraphs that if we forbid either of them in Hn(3), then the hypergraph is sparse, i.e. the number of its edges is o(n2). Since both contain a wicket, it implies a conjecture of Gyárfás and Sárközy that wicket-free hypergraphs are sparse.

Transference for loose Hamilton cycles in random 3‐uniform hypergraphs

AbstractA loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum ‐degree condition guarantees the existence of a loose Hamilton cycle in a ‐uniform hypergraph. For and each , the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of , our findings are asymptotically optimal.

Polynomial reconstruction problem for hypergraphs

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its “polynomial deck”, the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the “polynomial reconstruction problem” for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of 3-uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remains unresolved for ordinary graphs.

The hitting time of clique factors

AbstractIn [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let and let be divisible by . Then, in the random ‐uniform hypergraph process on vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph , the random graph process contains a ‐factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the ‐uniform hypergraph process ().

Linear saturation numbers of Berge-C3 and Berge-C4

The linear saturation number satklin(n,F) (linear extremal number exklin(n,F)) of F is the minimum (maximum) number of hyperedges of an n-vertex linear k-uniform hypergraph containing no member of F as a subgraph, but the addition of any new hyperedge such that the result hypergraph is still a linear k-uniform hypergraph creates a copy of some hypergraph in F. Determining ex3lin(n, Berge-C3) is equivalent to the famous (6,3)-problem, which has been settled in 1976. Since then, determining the linear extremal numbers of Berge cycles was extensively studied. As the counterpart of this problem in saturation problems, the problem of determining the linear saturation numbers of Berge cycles is considered. In this paper, we prove that satklin(n, Berge-Ct)≥⌊n−1k−1⌋ for any integers k≥3, t≥3, and the equality holds if t=3. In addition, we provide an upper bound for sat3lin(n, Berge-C4) and for any disconnected Berge-C4-saturated linear 3-uniform hypergraph, we give a lower bound for the number of hyperedges of it.

An Invitation to Higher Arity Science

Analytical thinking is dominated by binary ideas. From pairwise interactions to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this paper, we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this, we discuss several higher-order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.

3-Uniform hypergraphs from vector spaces

The Fundamental Theorem of Projective Geometry states that, in a vector space, a permutation of vector lines preserving triples that span a vector plane is induced by a semi-linear automorphism. We consider a generalisation to triples of subspaces, not necessarily of the same dimension, spanning, or being contained in a subspace of fixed dimension. We determine all cases in which the permutation is necessarily induced by a semi-linear automorphism.

Degree-based indices of hypergraphs: Definitions and first results

For a [Formula: see text]-uniform hypergraph [Formula: see text], we introduce degree-based indices such as the general sum-connectivity index [Formula: see text] and the general Randić index [Formula: see text], where [Formula: see text], [Formula: see text] is the set of hyperedges of [Formula: see text] and [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]; [Formula: see text] and [Formula: see text]. Other indices such as the first and second Zagreb index, first and second hyper-Zagreb index, classical sum-connectivity index, classical Randić index and harmonic index of a hypergraph [Formula: see text] are special cases of the general indices. For [Formula: see text], we obtain upper bounds on [Formula: see text] and [Formula: see text] for a uniform hypergraph [Formula: see text] with given order, order and number of isolated vertices, order and maximum degree, order and diameter at least [Formula: see text], and lower bounds for uniform hypergraphs with given order and no isolated vertices, order and minimum degree, and order and maximum possible degree. We also present extremal graphs for all the bounds. Bounds on Zagreb indices follow from our results on general indices.

Large monochromatic components in expansive hypergraphs

AbstractA result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$ . We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$ -uniform hypergraph on $n$ vertices (that is, a $k$ -uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$ ), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$ ). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$ -partite $r$ -uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$ , there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$ , then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$ ). We prove our results in this dual setting.

Some Discrete Tomography Problems in Hypergraph Model Interpretation

Abstract In this paper we consider discrete tomography problems with an additional requirement of non-repeatability of rows of the binary matrix to be reconstructed; as well as discrete tomography problems with given pairwise projections. Representing the problems in the hypergraph model and pointing out their equivalence to the basic definitions, we state the following results: (i) nonconvexity of the set of hypergraphic sequences of simple hypergraphs with $$n$$ vertices and $$m$$ hyperedges in the $$n$$-dimensional $$m + 1$$-valued lattice, (ii) characterization of monotone Boolean functions associated with degree sequences of 3-/(n–3)-uniform hypergraphs, (iii) formulation of discrete tomography problems with paired projections, their connection to hypergraph degree sequence problem with generalized degrees, a solution for a particular case.

A family of 4-uniform hypergraphs with no stability

A family of 4-uniform hypergraphs with no stability

Regarding equitable colorability defect of hypergraphs

After Lovász’s break-through in determining the chromatic number of Kneser graphs (1978), and after extending this result to the chromatic number of $r$-uniform Kneser hypergraphs by Alon, Frankl, and Lovász’s (1986), some important parameters such as colorability defect and equitable colorability defect were introduced in order to provide sharp lower bounds for the chromatic number of general $r$-uniform Kneser hypergraphs. As a generalization of many earlier results in this area, Azarpendar and Jafari (2023) introduced the $s$-th equitable $r$-colorability defect ${\rm ecd}^r (\mathcal{F} , s)$; a parameter which provides a lower bound for the chromatic number of generalized Kneser hypergraphs ${\rm KG} ^r (\mathcal{F} , s)$. They proved the following nice inequality $$\chi \left( {\rm KG} ^r (\mathcal{F} , s) \right) \geq \left\lceil \frac{ {\rm ecd}^r \left( \mathcal{F} , \left\lfloor \frac{s}{2} \right\rfloor \right) }{r-1} \right\rceil ,$$ and noted that it is plausible that the above inequality remains true if one replaces $\left\lfloor \frac{s}{2} \right\rfloor$ with $s$. In this paper, considering the relation ${\rm ecd}^r \left( \mathcal{F} , x \right) \geq {\rm cd}^r \left( \mathcal{F} , x \right)$ which always holds, we show that even in the weaker inequality $$\chi \left( {\rm KG} ^r (\mathcal{F} , s) \right) \geq \left\lceil \frac{ {\rm cd}^r \left( \mathcal{F} , \left\lfloor \frac{s}{2} \right\rfloor \right) }{r-1} \right\rceil ,$$ no number $x$ greater than $\left\lfloor \frac{s}{2} \right\rfloor$ could be replaced by $\left\lfloor \frac{s}{2} \right\rfloor$.