Class Of Hypergraphs Research Articles

On positive local combinatorial dividing-lines in model theory

We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

Connectivity in Hypergraphs

AbstractIn this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

Axiomatisability and hardness for universal Horn classes of hypergraphs

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs.

Coloring Cross-Intersecting Families

Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erdős-Ko-Rado theorem. On the other hand, P. Erdős and L. Lovász in their noted 1975 paper posed problems on coloring intersecting families as a restriction of classical hypergraph coloring problems to a special class of hypergraphs. This note deals with the mentioned coloring problems stated for cross-intersecting families.

Prime splittings of determinantal ideals

ABSTRACTWe consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gröbner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That is, we write the determinantal ideal as a sum of smaller determinantal ideals such that each minimal prime is a sum of minimal primes of the summands.

Chromatic Symmetric Functions of Hypertrees

The chromatic symmetric function $X_H$ of a hypergraph $H$ is the sum of all monomials corresponding to proper colorings of $H$. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental quasisymmetric functions $F_S$, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs $H$ — hypertrees with prime-sized edges — for which $X_H$ is $F$-positive, and give an explicit combinatorial interpretation for the $F$-coefficients of $X_H$.

Stability and Turán Numbers of a Class of Hypergraphs via Lagrangians

Given a family ofr-uniform hypergraphs${\cal F}$(orr-graphs for brevity), the Turán number ex(n,${\cal F})$of${\cal F}$is the maximum number of edges in anr-graph onnvertices that does not contain any member of${\cal F}$. A pair {u,v} iscoveredin a hypergraphGif some edge ofGcontains {u, v}. Given anr-graphFand a positive integerp⩾n(F), wheren(F) denotes the number of vertices inF, letHFpdenote ther-graph obtained as follows. Label the vertices ofFasv1,. . .,vn(F). Add new verticesvn(F)+1,. . .,vp. For each pair of verticesvi, vjnot covered inF, add a setBi,jofr− 2 new vertices and the edge {vi, vj} ∪Bi,j, where theBi,jare pairwise disjoint over all such pairs {i, j}. We callHFpthe expanded p-clique with an embedded F. For a relatively large family ofF, we show that for all sufficiently largen, ex(n,HFp) = |Tr(n, p− 1)|, whereTr(n, p− 1) is the balanced complete (p− 1)-partiter-graph onnvertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for largen).

Learning a hidden uniform hypergraph

Motivated by applications in genome sequencing, Grebinski and Kucherov (Discret Appl Math 88:147–165, 1998) studied the graph learning problem which is to identify a hidden graph drawn from a given class of graphs with vertex set \(\{1,2,\ldots ,n\}\) by edge-detecting queries. Each query tells whether a set of vertices induces any edge of the hidden graph or not. For the class of all hypergraphs whose edges have size at most r, Chodoriwsky and Moura (Theor Comput Sci 592:1–8, 2015) provided an adaptive algorithm that learns the class in \(O(m^r\log n)\) queries if the hidden graph has m edges. In this paper, we provide an adaptive algorithm that learns the class of all r-uniform hypergraphs in \(mr\log n+(6e)^rm^{\frac{r+1}{2}}\) queries if the hidden graph has m edges.

Super edge-magic labeling of [formula omitted]-node [formula omitted]-uniform hyperpaths and [formula omitted]-node [formula omitted]-uniform hypercycles

We generalize the notion of the super edge-magic labeling of graphs to the notion of the super edge-magic labeling of hypergraphs. For a hypergraph H with a finite vertex set V and a hyperedge set E, a bijective function f:V∪E→{1,2,3,…,|V|+|E|} is called a super edge-magic labeling if it satisfies (i) there exists a magic constant Λ such that f(e)+∑v∈ef(v)=Λ for all e∈E and (ii) f(V)={1,2,3,…,|V|}. A hypergraph admitting a super edge-magic labeling is said to be super edge-magic. In this paper, we show the equivalent form of this labeling, i.e, a hypergraph H is super edge-magic if and only if there exists a bijective function f:V→{1,2,3,…,|V|} such that {∑v∈ef(v)|e∈E} is the set of |E| consecutive integers. Finally, we define two classes of hypergraphs, namely m-nodek-uniform hyperpaths and m-nodek-uniform hypercycles which are denoted by mPn(k) and mCn(k), respectively. We show that under some conditions the hypergraphs mPn(k) and mCn(k) are super edge-magic.

Edge-coloring of 3-uniform hypergraphs

We consider edge-colorings of 3-uniform hypergraphs which is a natural generalization of the problem of edge-colorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NP-complete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.

Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs. This class of hypergraphs corresponds to matrices with the circular ones property, which play an important role in computational genomics. Our results imply that there is a logspace algorithm that decides whether a given matrix has this property.Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.Note that solving a problem in logspace implies that it is solvable by a parallel algorithm of the class AC1. For the problems under consideration, at most AC2 algorithms were known earlier.

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $$c \left( {\begin{array}{c}|V(H)| r-1\end{array}}\right) $$c|V(H)|r-1 has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomass? and then solved by Allen, B?ttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turan density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobas---Erd?s graphs defined earlier by the authors.

Perfect Packings in Quasirandom Hypergraphs II

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties: •for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,•for every vertex x of H, the link graph of x is a quasirandom graph with density at least p. Then H has a perfect Kr(3)-packing. Moreover, we show that neither of the hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's Hypergraph Blow-up Lemma, with a slightly stronger hypothesis on H.

Nordhaus–Gaddum-type problems for lines in hypergraphs

We study the number of lines in hypergraphs in a symmetric setting, where both the hypergraph and its complement are considered. In the general case and in some special cases, the lower bounds on the number of lines are much higher than their counterparts in single hypergraph setting or admit more elegant proofs. We show that the minimum value of product of the number of lines in both hypergraphs on n points is easily determined as (n2); and the minimum value of their sum is between Ω(n) and O(nlogn). We also study some restricted classes of hypergraphs; and determine the tight bounds on the minimum sum when the hypergraph is derived from a Euclidean space, a real projective plane, or a tree.

On the Chromatic Thresholds of Hypergraphs

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

Some Results on Exclusive Sum Labelings of Hypergraphs

We generalize the concept of exclusive sum labelings of graphs (cf. Miller et al., JCMCC 55:137---148, 2005) and determine the exclusive sum number for several classes of hypergraphs. The results disclose a significant difference between graphs (i.e. 2-uniform hypergraphs) and "genuine" hypergraphs.

Threefold triple systems with nonsingular [formula omitted

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix N2 (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular N2 (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near (v2) is obtained for large orders v.

Редукционные методы восстановления некоторого класса гиперграфов

Редукционные методы восстановления некоторого класса гиперграфов

Hypergraph recovery algorithms from a given vector of vertex degrees

Methods of obtaining certain classes of hypergraphs from a given integer vector of vertex degrees are considered. These classes are as follows: hyperedges with unit weight incident upon k vertices; hyperedges with unit weight incident upon k vertices in the case when the vertices may be non-unique; multiple hyperedges incident upon k vertices; and arbitrary hypergraph in which the edges can contain any set of k vertices. For each of these classes, an algorithm is proposed for constructing the hypergraph from an arbitrary vector. If the construction is impossible, the algorithm determines how much the vector should be reduced so that the hypergraph could be constructed.

Structural Tractability of Counting of Solutions to Conjunctive Queries

We explore the complexity of counting solutions to conjunctive queries, a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query ϕ, which measures how the free variables are spread in ϕ. As usual in database theory, we associate a hypergraph to a query ϕ. We show that for classes of queries for which these associated hypergraphs admit good decompositions, e.g., bounded width generalized hypertree decompositions, bounded quantified star size exactly characterizes the subclasses of hypergraphs for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of hypergraphs for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a conjunctive query.