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The extremal number (,) is the maximum number of edges in an -vertex graph containing no subgraph isomorphic to . is the complete graph on vertices. (,) is the Turán graph: a complete -partite graph on vertices, with vertices distributed between parts as equally as possible.
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 Paul Erdős and Arthur Stone, who proved it in 1946, [1] and it has been described as the “fundamental theorem of extremal graph theory”. [2]
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 extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that ...
The Turán graph T(n,r) is an example of an extremal graph. It has the maximum possible number of edges for a graph on n vertices without (r + 1)-cliques. This is T(13,4). 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 ...
A balanced tripartite graph with the unique triangle property can be made into a partitioned bipartite graph by removing one of its three subsets of vertices, and making an induced matching on the neighbors of each removed vertex. To convert a graph with a unique triangle per edge into a triple system, let the triples be the triangles of the graph.
Comparability graphs: Induced subgraph Triangle-free graphs: Triangle K 3: Induced subgraph Definition Planar graphs: K 5 and K 3,3: Homeomorphic subgraph Kuratowski's theorem: K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph ...
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular. The lemma shows that certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs.
Packing and covering triangles in the complete graph. The maximum number of edge-disjoint triangles in this graph is two (left). If four edges are removed from the graph (red edges, right), the remaining subgraph becomes triangle-free, and more strongly bipartite (as shown by the blue and yellow vertex coloring). According to Tuza's conjecture ...
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