Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Lemme de régularité de Szemerédi
Résumé
Szemerédi's regularity lemma is one of the most powerful tools in extremal graph theory, particularly in the study of large dense graphs. It states that the vertices of every large enough graph can be partitioned into a bounded number of parts so that the edges between different parts behave almost randomly.
According to the lemma, no matter how large a graph is, we can approximate it with the edge densities between a bounded number of parts. Between any two parts, the distribution of edges will be pseudorandom as per the edge density. These approximations provide essentially correct values for various properties of the graph, such as the number of embedded copies of a given subgraph or the number of edge deletions required to remove all copies of some subgraph.
To state Szemerédi's regularity lemma formally, we must formalize what the edge distribution between parts behaving 'almost randomly' really means. By 'almost random', we're referring to a notion called ε-regularity. To understand what this means, we first state some definitions. In what follows G is a graph with vertex set V.
Definition 1. Let X, Y be disjoint subsets of V. The edge density of the pair (X, Y) is defined as:
where E(X, Y) denotes the set of edges having one end vertex in X and one in Y.
We call a pair of parts ε-regular if, whenever you take a large subset of each part, their edge density isn't too far off the edge density of the pair of parts. Formally,
Definition 2. For ε > 0, a pair of vertex sets X and Y is called ε-regular, if for all subsets A ⊆ X, B ⊆ Y satisfying |A| ≥ ε|X|, |B| ≥ ε|Y|, we have
The natural way to define an ε-regular partition should be one where each pair of parts is ε-regular. However, some graphs, such as the half graphs, require many pairs of partitions (but a small fraction of all pairs) to be irregular. So we shall define ε-regular partitions to be one where most pairs of parts are ε-regular.
Definition 3. A partition of into sets is called an -regular partition if
Now we can state the lemma:
Szemerédi's Regularity Lemma.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.