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Hypergraphs are systems of finite sets and form, probably, the most general concept in discrete mathematics. This branch of mathematics has developed very rapidly during the latter part of the twentieth century, influenced by the advent of computer science. Many theorems on set systems were already known at the beginning of the twentieth century, but these results did not form a mathematical field in itself. It was only in the early 1960s that hypergraphs become an independent theory. Hence, hypergraph theory is a recent theory. It was mostly developed in Hungary and France under the leadership of mathematicians like Paul Erdös, László Lovász, Paul Turán,… but also by C. Berge, for the French school. Originally, developed in France by Claude Berge in 1960, it is a generalization of graph theory. The basic idea consists in considering sets as generalized edges and then in calling hypergraph the family of these edges (hyperedges). As extension of graphs, many results on trees, cycles, coverings, and colorings of hypergraphs will be seen in this book.
Hypergraphs model more general types of relations than graphs do. In the past decades, the theory of hypergraphs has proved to be of a major interest in applications to real-world problems. These mathematical tools can be used to model networks, biology networks, data structures, process scheduling, computations and a variety of other systems where complex relationships between the objects in the system play a dominant role. From a theoretical point of view, hypergraphs allow to generalize certain theorems on graphs, even to replace several theorems on graphs by a single theorem of hypergraphs. For instance, the Berge’s weak perfect graph conjecture, which says that a graph is perfect if and only if its complement is perfect, was proved thanks to the concept of normal hypergraph. From a practical point of view, they are now increasingly preferred to graphs.
In this book, we give a general and nonstandard presentation of the theory of hypergraphs, although many paragraphs deal with the traditional elements of this theory.
Hypergraphs: Basic Concepts
Hypergraphs: First Properties
Hypergraph Colorings
Some Particular Hypergraphs
Reduction-Contraction of Hypergraph
Dirhypergraphs: Basic Concepts
Applications of Hypergraph Theory: A Brief Overview