The Regularity Method for Sparse Graphs and Hypergraphs

稀疏图和超图的正则方法

• 批准号: 5443326
• 负责人: Professor Dr. Mathias Schacht
• 金额: --
• 依托单位: Forschungsschwerpunkt Diskrete Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2007-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5443326?language=en
• 关键词: Regularity; Method; Sparse; Graphs; Hypergraphs

This proposal seeks further development and understanding of the Regularity Method, which has had a series of successes in Discrete Mathematics. The Regularity Lemma of E. Szemerédi for graphs asserts that every dense graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type. This observation is called Counting Lemma. The interplay of Szemerédi's Regularity Lemma and the Counting Lemma, referred to as the Regularity Method for dense graphs, has many applications in the area ot extremal graph theory. Since random graphs of a given edge-density are usually easier to analyze than arbitrary graphs of the same edge-density, the Regularity Method enables us to transfer some methods applicable to random graphs to the class of all graphs. In recent years the Regularity Method was partly expanded to new types of discrete structures: sparse graphs and k-uniform hypergraphs. In particular, analogues of the Regularity Lemma for these combinatorial objects were established. In the proposed program, we seek a deeper understanding of the random-like behavior guaranteed by those Regularity Lemmas. Furthermore, we focus on applications of these novel techniques in the area of extremal combinatorics.

这一建议寻求对在离散数学中取得一系列成功的正则性方法的进一步发展和理解。E.Szemerédi关于图的正则性引理认为，每个稠密图都可以分解成相对较少的类随机子图。这种类似随机的行为使人们能够找到并枚举给定同构类型的子图。这一观察结果称为计数引理。Szemerédi正则性引理和Counting引理的相互作用，被称为稠密图的正则性方法，在极值图论领域有着广泛的应用。由于具有给定边密度的随机图通常比具有相同边密度的任意图更容易分析，因此正则性方法使我们能够将一些适用于随机图的方法转移到所有图的类。近年来，正则性方法被部分地扩展到新的离散结构：稀疏图和k-一致超图。特别地，建立了这些组合对象的正则性引理的类比。在所提出的程序中，我们寻求对那些正则性引理所保证的类随机行为的更深层次的理解。此外，我们还重点讨论了这些新技术在极值组合数学领域中的应用。