Number Of Hypergraphs Research Articles

Chromatic numbers of hypergraphs

Chromatic numbers of hypergraphs

Linear Turán numbers of acyclic triple systems

The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F, the linear Turán number exL(n,F) is the maximum number of triples in a linear triple system with n points that does not contain F as a subsystem.We initiate the study of the linear Turán number for an acyclicF. In this case exL(n,F) is linear in n and we aim for good bounds. Since the case of trees is already difficult for graphs (Erdős–Sós conjecture), we concentrate on matchings, paths and small trees.In case of matchings, where Mk is the set of k pairwise disjoint triples, we prove that for fixed k and large enough n, exL(n,Mk)=f(n,k) where f(n,k) is the maximum number of triples that can meet k−1 points in a linear triple system on n points. This is an analogue of an old result of Erdős on hypergraph matchings. For the k-edge linear path Pk we show (extending some standard path increasing methods used for graphs) that exL(n,Pk)≤1.5kn which is probably far from best possible.Finding exL(n,F) relates to difficult problems on Steiner triple systems and interesting even for small trees. For example, for P4, the path with four triples, exL(n,P4)≤4n3 with equality only for disjoint union of affine planes of order 3. On the other hand, for E4, the tree having three pairwise disjoint triples and a fourth one meeting all of them, we have bounds only: 6⌊n−34⌋≤exL(n,E4)≤2n.

The number of hypergraphs without linear cycles

The r-uniform linear k-cycle Ckr is the r-uniform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers r,k≥3, the number of Ckr-free r-uniform hypergraphs on n vertices is 2Θ(nr−1), thereby settling a conjecture due to Mubayi and Wang from 2017.

New Lower Bounds for Secure Codes and Related Hash Families: A Hypergraph Theoretical Approach

Various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials. The codewords in such a structure can be regarded as a subset of $Q^{N}$ , the set of all $q$ -ary vectors of given length $N$ , satisfying some constraints. We use a hypergraph model to characterize the combinatorial structure. By applying a result of Duke et al. on the lower bound of the independence number of hypergraphs, we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families. In particular, the general method is illustrated via the examples of existence results on some perfect hash families, frameproof codes, and separable codes.

On the Size‐Ramsey Number of Hypergraphs

AbstractThe size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs.We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erd\H{o}s, Hajnal, Simonovits, S\'os, and Szemer\'edi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = \Omega(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) = \Omega(n^2)$; our constructions also imply several results on the Ramsey-Tur\'an numbers of hypergraphs.

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.

The circular chromatic number of hypergraphs

A generalization of the circular chromatic number to hypergraphs is discussed. In particular, it is indicated how the basic theory, and five equivalent formulations of the circular chromatic number of graphs, can be carried over to hypergraphs with essentially the same proofs.

The genus of G-spaces and topological lower bounds for chromatic numbers of hypergraphs

Lower bounds for chromatic numbers of hypergraphs via the genus and cohomological index of special complexes (Hom-and JHom-complexes) are given.

Extensions of the linear bound in the Füredi–Hajnal conjecture

We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1-entries in an n × n ( 0 , 1 ) -matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Brändén and Mansour. We then extend the original Füredi–Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1-entries in a d-dimensional ( 0 , 1 ) -matrix with side length n which avoids a d-dimensional permutation matrix is O ( n d − 1 ) .

The cutwidth and the vertex separation number of hypergraphs and their König’s representations

The cutwidth and the vertex separation number of hypergraphs and their König’s representations

On the cyclomatic number of a hypergraph

This note generalizes the notion of cyclomatic number (or cycle rank) from Graph Theory to Hypergraph Theory and links it up with the concept of planarity in hypergraphs which was recently introduced by R.P. Jones. Sharp bounds are obtained for the cyclomatic number of the planar hypergraphs and, further, it is shown that the upper bound is attainable if, and only if the hypergraph satisfies Krewera's condition.