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The Turán graph T ( 2n, n ) can be formed by removing a perfect matching from a complete graph K < sub > 2n </ sub >.
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Turán and graph
The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory.
The Turán graph T ( n, r ) is a graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets.
Turán graphs are named after Pál Turán, who used them to prove Turán's theorem, an important result in extremal graph theory.
By the pigeonhole principle, any set of r + 1 vertices in the Turán graph includes two vertices in the same partition subset ; therefore,
the Turán graph does not contain a clique of size r + 1.
According to Turán's theorem, the Turán graph has the maximum possible number of edges among all ( r + 1 )- clique-free graphs with n vertices.
Keevash and Sudakov ( 2003 ) show that the Turán graph is also the only ( r + 1 )- clique-free graph of order n in which every subset of αn vertices spans at least edges, if α is sufficiently close to 1.
The Erdős – Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph.
The octahedron, a cross polytope whose edges and vertices form a Turán graph T ( 6, 3 ).
Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied.
The Turán graph T ( n, 2 ) is a complete bipartite graph and, when n is even, a Moore graph.
When r is a divisor of n, the Turán graph is symmetric and strongly regular, although some authors consider Turán graphs to be a trivial case of strong regularity and therefore exclude them from the definition of a strongly regular graph.
The Turán graph has 3 < sup > a </ sup > 2 < sup > b </ sup > maximal cliques, where
Turán and T
It was open for some time whether T ( n ) ≥ 0 for sufficiently big n ≥ n < sub > 0 </ sub > ( this " conjecture " is occasionally ( but incorrectly ) attributed to Pál Turán ).
An n-vertex graph G is a subgraph of a Turán graph T ( n, r ) if and only if G admits an equitable coloring with r colors.
We call the resulting graph the Turán graph T ( n, r ).
The Turán graph T ( 13, 4 ), an example of a cograph
Turán and n
Witsenhausen ( 1974 ) conjectures that the maximum sum of squared distances, among n points with unit diameter in R < sup > d </ sup >, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.
Extremal graph theory started in 1941 when Turán proved his theorem determining those graphs of order n, not containing the complete graph K < sub > k </ sub > of order k, and extremal with respect to size ( that is, with as many edges as possible ).
In 1934 Turán gave a new and very simple proof of a 1917 result of G. H. Hardy and Ramanujan on the normal order of the number of distinct prime divisors of a number n, namely that it is very close to ln ln n. In probabilistic terms he estimated the variance from ln ln n. Halász says " Its true significance lies in the fact that it was the starting point of probabilistic number theory ".
Turán and can
Every Turán graph is a cograph ; that is, it can be formed from individual vertices by a sequence of disjoint union and complement operations.
Specifically, such a sequence can begin by forming each of the independent sets of the Turán graph as a disjoint union of isolated vertices.
He is also known for the Kövari – Sós – Turán theorem bounding the number of edges that can exist in a bipartite graph with certain forbidden subgraphs,
Turán and be
In number theory, Szemerédi's theorem is a result that was formerly the Erdős – Turán conjecture ( not to be confused with Erdős – Turán conjecture on additive bases ).
Turán and by
A confirmation of this positivity conjecture would have led to a proof of the Riemann hypothesis, as was shown by Pál Turán.
Falls, Powell, and Snoeyink develop an efficient algorithm for finding clusters of orthologous groups of genes in genome data, by representing the data as a graph and searching for large Turán subgraphs.
Turán graphs were first described and studied by Hungarian mathematician Paul Turán in 1941, though a special case of the theorem was stated earlier by Mantel in 1907.
The problem of determining the number was suggested by Paul Turán and became known as Turán's brick factory problem.
* The life of Alfréd Rényi, by Pál Turán
A quantitative form of the Weyl criterion is given by the Erdős – Turán inequality.
Turán and from
He is best known for his proof from 1975 of an old conjecture of Paul Erdős and Paul Turán: if a sequence of natural numbers has positive upper density then it contains arbitrarily long arithmetic progressions.
Turan ( Hungarian: Turán ) derives from the Persian توران and refers to the steppes of Central Asia, land of the Tur.
Turán and complete
He invented the Turán graph, a generalization of the complete bipartite graph, to prove his theorem.
All complete graphs, complete bipartite graphs, threshold graphs, and Turán graphs are cographs.
Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs, and their complements, the Turán graphs.
Turán and K
Turán's theorem states that the Turán graph has the largest number of edges among all K < sub > r + 1 </ sub >- free n-vertex graphs.
The graphs meeting this bound are the Moon – Moser graphs K < sub > 3, 3 ,...</ sub >, a special case of the Turán graphs arising as the extremal cases in Turán's theorem.
Turán also found the ( unique ) largest graph not containing K < sub > k </ sub > which is named after him, namely Turán graph.
Turán and <
In 1936 Erdős and Turán conjectured for every value d called density 0 < d < 1 and every integer k there is a number N ( d, k ) such that every subset A of
Turán and >
Pór and Wood ( 2005 ) give a lower bound of Ω (( rn )< sup > 3 / 4 </ sup >) on the volume of any three-dimensional grid embedding of the Turán graph.
graph and T
Each point with abscissa T on the graph represents an intersection between C and Af.
The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).
The pioneering work of W. T. Tutte was very influential in the subject of graph drawing.
The graph of a monotone operator G ( T ) is a monotone set.
Let T be a subset of the vertex set of a graph.
( In a connected graph, a T-join exists if and only if | T | is even.
The snark theorem, a result conjectured by W. T. Tutte and announced in 2001 by Robertson, Sanders, Seymour, and Thomas, states that every snark has the Petersen graph as a minor.
However, boundedness of the inverse does follow directly from its existence if one introduces the additional assumption that T is closed ; this follows from the closed graph theorem.
A net is a bipartite graph, where P is one partition and T is the other.
* Graphviz, an open-source graph drawing system from AT & T
call it “ one of the deepest unsolved problems in graph theory .” Another result relating the four-color theorem to graph minors is the snark theorem announced by Robertson, Sanders, Seymour, and Thomas, a strengthening of the four-color theorem conjectured by W. T. Tutte and stating that any bridgeless 3-regular graph that requires four colors in an edge coloring must have the Petersen graph as a minor.
This graph is also the 1-skeleton of an n-dimensional cross-polytope ; for instance, the graph T ( 6, 3 ) = K < sub > 2, 2, 2 </ sub > is the octahedral graph, the graph of the regular octahedron.
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