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Concept
Turán's theorem
Summary
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 does not have a given subgraph.
An example of an -vertex graph that does not contain any -vertex clique may be formed by partitioning the set of vertices into parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph . Turán's theorem states that the Turán graph has the largest number of edges among all Kr+1-free n-vertex graphs.
Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. The special case of the theorem for triangle-free graphs is known as Mantel's theorem; it was stated in 1907 by Willem Mantel, a Dutch mathematician.
Turán's theorem states that every graph with vertices that does not contain as a subgraph has at most as many edges as the Turán graph . For a fixed value of , this graph hasedges, using little-o notation. Intuitively, this means that as gets larger, the fraction of edges included in gets closer and closer to . Many of the following proofs only give the upper bound of .
list five different proofs of Turán's theorem.
Many of the proofs involve reducing to the case where the graph is a complete multipartite graph, and showing that the number of edges is maximized when there are parts of size as close as possible to equal.
This was Turán's original proof. Take a -free graph on vertices with the maximal number of edges. Find a (which exists by maximality), and partition the vertices into the set of the vertices in the and the set of the other vertices.
Now, one can bound edges above as follows:
There are exactly edges within .
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