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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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Citations
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Three years of graphs and music : some results in graph theory and its applications
Nathann Cohen
- 20 Oct 2011
TL;DR: This thesis asserts that similarly to what Wilson's theorem implies for non-induced graph decomposition, there exists for any graph $H$ an infinite sequence of dense graph whose edge set can be partitioned in induced copies of $H$.
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Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
TL;DR: It is shown that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product, which coincide with the usual strong graph product whenever H is a graph.
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A generic disjunctive/conjunctive decomposition model for n -ary relations
TL;DR: In this paper, a generic decomposition model that represents an arbitrary n-ary relation as a disjunctive or conjunctive combination of a number of nary component relations of a prespecified type is presented.
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•Posted Content
Large homogeneous submatrices
TL;DR: It is proved that if an $n\times n$ zero-one matrix A does not contain P as a submatrix, then A has an $cn\times cn$ homogeneous sub matrix for a suitable constant $c>0$.
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Visualization of network structure by the application of hypernodes
TL;DR: This study shows that when networks are embedded in the geographical space hypernodes may relate to clusters in the spatial domain and the selection of the visual variables to illustrate the strength of the edges and nodes in a weighted network is discussed.
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