Open AccessBook
Hypergraphs: Combinatorics of Finite Sets
Claude Berge
- 11 Jul 2011
1K
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.
read more
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.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Posted Content
Making a tournament indecomposable by one subtournament-reversal operation
TL;DR: In this paper, it was shown that if the tournament is not transitive of even order, then it can be made indecomposable by reversing the arcs of a subtournament of the tournament.
Phylogenetic decisiveness and no-rainbow hypergraph coloring
Ghazaleh Parvini
- 01 Jan 2021
TL;DR: In this article, the authors propose a method to solve the problem of "uniformity" and "uncertainty" in the context of health care, and propose a solution.
Spectral aspects of mining complex networks
Shamamedova Zinnat Xayrulloyevna
- 24 Aug 2022
TL;DR: In this paper , the authors consider the geometric structure of complex networks at many levels, including the intrinsic metric structure of a graph using the theory of length spectrum and its relation to the eigenvalues of the non-backtracking matrix of the graph.
Partial latin squares and their generalized quotients
L. Yu. Glebsky,Carlos J. Rubio +1 more
TL;DR: In this article, a necessary and sufficient condition for a set of triples to be a quotient of a (partial) latin square is given. But the condition is not defined.
The structure of co-publications multilayer network
TL;DR: In this article, a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers is proposed, where each clique is built from new and existing nodes selected using preferential attachment.