Hypergraphs - Combinatorics of Finite Sets

Abstract

1. General Concepts. Dual Hypergraphs. Degrees. Intersecting Families. The Coloured Edge Property and Chvatal'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 Konig 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. References.