Hypergraphs, Vol 45. North-Holland Mathematical Library

  • ID: 1766308
  • Book
  • 254 Pages
  • Elsevier Science and Technology
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Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. The idea of looking at a family of sets from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.

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1. General Concepts. Dual Hypergraphs. Degrees. Intersecting Families. The Coloured Edge Property and Chvátal's Conjecture. The Helly Property. Section of a Hypergraph and the Kruskal-Katona Theorem. Conformal Hypergraphs. Representative Graphs.

2. Transversal Sets and Matchings. Transversal Hypergraphs. The Coefficients r and r'. r-Critical Hypergraphs. The König Property.

3. Fractional Transversals. Fractional Transversal Number. Fractional Matching of a Graph. Fractional Transversal Number of a Regularisable Hypergraph. Greedy Transversal Number. Ryser's Conjecture. Transversal Number of Product Hypergraphs.

4. Colourings. Chromatic Number. Particular Kinds of Colourings. Uniform Colourings. Extremal Problems Related to the Chromatic Number. Good Edge-Colourings of a Complete Hypergraph. An Application to an Extremal Problem. Kneser's Problem.

5. Hypergraphs Generalising Bipartite Graphs. Hypergraphs without Odd Cycles. Unimodular Hypergraphs. Balanced Hypergraphs. Arboreal Hypergraphs. Normal Hypergraphs. Mengerian Hypergraphs. Paranormal Hypergraphs.

Appendix: Matchings and Colourings in Matroids.

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Berge, C.
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