Category: Extremal graph theory

Ore's theorem
Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore. It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with su
Erdős–Stone theorem
In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a non-complete graph H. It is named after Pau
Turán graph
The Turán graph, denoted by , is a complete multipartite graph; it is formed by partitioning a set of vertices into subsets, with sizes as equal as possible, and then connecting two vertices by an edg
Extremal graph theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies h
Even circuit theorem
In extremal graph theory, the even circuit theorem is a result of Paul Erdős according to which an n-vertex graph that does not have a simple cycle of length 2k can only have O(n1 + 1/k) edges. For in
Turán's theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extr
Ruzsa–Szemerédi problem
In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle.Eq
Zarankiewicz problem
The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgr
Container method
The method of (hypergraph) containers is a powerful tool that can help characterize the typical structure and/or answer extremal questions about families of discrete objects with a prescribed set of l
Common graph
In graph theory, an area of mathematics, common graphs belong to a branch of extremal graph theory concerning inequalities in homomorphism densities. Roughly speaking, is a common graph if it "commonl
Forbidden subgraph problem
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges an -vertex graph can have such that it does not have a subgraph isom
Biclique-free graph
In graph theory, a branch of mathematics, a t-biclique-free graph is a graph that has no 2t-vertex complete bipartite graph Kt,t as a subgraph. A family of graphs is biclique-free if there exists a nu
Dependent random choice
In mathematics, dependent random choice is a probabilistic technique that shows how to find a large set of vertices in a dense graph such that every small subset of vertices has many common neighbors.
Turán number
In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determ
Homomorphism density
In the mathematical field of extremal graph theory, homomorphism density with respect to a graph is a parameter that is associated to each graph in the following manner: . Above, is the set of graph h