Cartesian product of hypergraphs: properties and algorithms

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspects of Cartesian products of hypergraphs. We also extend a classical prime factorization algorithm initially designed for graphs to connected conformal hypergraphs using 2-sections of hypergraphs

Covering graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi

The magnitude homology of a hypergraph

The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with higher-order structures. In this paper, we focus on describing the topological characteristics of hypergraphs by considering their magnitude homology. We begin by examining the distances between hyperedges in a hypergraph and establish the magnitude homology of hypergraphs. Additionally, we explore the relationship between the magnitude and the magnitude homology of hypergraphs. Furthermore, we derive several functorial properties of the magnitude homology for hypergraphs. Lastly, we present the K\"{u}nneth theorem for the simple magnitude homology of hypergraphs

Hypergraph Products

In this work, new definitions of hypergraph products are presented. The main focus is on the generalization of the commutative standard graph products: the Cartesian, the direct and the strong graph product. We will generalize these well-known graph products to products of hypergraphs and show several properties like associativity, commutativity and distributivity w.r.t. the disjoint union of hypergraphs. Moreover, we show that all defined products of simple (hyper)graphs result in a simple (hyper)graph. We will see, for what kind of product the projections into the factors are (at least weak) homomorphisms and for which products there are similar connections between the hypergraph products as there are for graphs. Last, we give a new and more constructive proof for the uniqueness of prime factorization w.r.t. the Cartesian product than in [Studia Sci. Math. Hungar. 2: 285–290 (1967)] and moreover, a product relation according to such a decomposition. That might help to find efficient algorithms for the decomposition of hypergraphs w.r.t. the Cartesian product