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Hypergraphs: Combinatorics of Finite Sets
Claude Berge
- 11 Jul 2011
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TL;DR: This chapter discusses Hypergraphs Generalising Bipartite Graphs, which are a collection of hypergraphs designed to solve the problem of Uniform Colourings in Matroids, and some of the properties of these graphs.
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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.
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Loïc Foissy
- 03 Apr 2023
TL;DR: In this paper , the authors introduce two structures of cointeracting bialgebras on hypergraphs, induced by two notions of induced sub-hypergraphs and obtain Hopf-algebraic proves about the values of this chromatic polynomial in -1 or about its coefficients, with the help of the action of a monoid of characters.
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An edge colouring of multigraphs.
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TL;DR: The maximum number of colours in a strict edge colouring of G is called the upper chromatic index of G and is denoted by χ(G) and is proved in this paper.
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