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Topic: Turan graph

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Extremal graph theory

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	NationMaster - Encyclopedia: Turan graph (Site not responding. Last check: 2007-10-15)
		The Turán graph T(n,r) is the complete r-partite graph with n vertices whose partite sets differ in size at most 1.
		Graph theory In graph theory, TurÃ¡ns theorem is a result on the number of edges in a Ks+1-free graph.
		Extremal graph theory is a branch of mathematics.
	www.nationmaster.com /encyclopedia/Turan-graph (345 words)

	Turan graph - Wikipedia, the free encyclopedia
		The Turán graph T(n,r) is the complete r-partite graph with n vertices whose partite sets differ in size at most 1.
		Clearly, the Turán graph T(n,r) does not contain a clique of size r+1.
		This is the best possible (with respect to the number of edges) among all graphs with n vertices (see Turán's theorem).
	en.wikipedia.org /wiki/Turan_graph (162 words)

	ACO Seminar, 2005-2006
		Given two graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. We allow partitions only, that is, every edge of G appears in precisely one part.
		For a non-empty graph H we denote by gcd(H) the greatest common divisor of the degrees of H. For a bipartite graph with gcd(H)=1 the exact value of the function $\phi_H(n)$ will be obtained, for n sufficiently large.
		The second model is a random graph on n vertices in which (1-b)n vertices have degree 4 and the other bn vertices have degree 5.
	www.cs.cmu.edu /~aco/seminar20052006.html (2772 words)

	Turan Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-15)
		In Geography, Turan refers the bulk of the Eurasian landmass including the Russian steppes, Central Asian Turkestan, Mongolia, the Caucasus and other regions where historical Hunnish, Avar, Turkic and Mongol powers held sway, such as Iran and Anatolia.
		This name was established in the Persian-Turkish literary tradition by Shahnameh, designating the Turkic hordes north of Iran, used in a sense contradictive from the latter.
		Turan was also the Hungarian tank of the WWII - a total of 424 made in two variants: Turan I with 40 mm gun and Turan II with 75 mm gun.
	www.borednet.com /e/n/encyclopedia/t/tu/turan.html (284 words)

	Summer School 2006 - Algorithms, Structure, Randomness
		The lectures given on extremal graph theory and random graphs will be supplemented by exercises which will be solved in small groups with solution sessions in the evenings.
		Extremal graph theory and Ramsey theory were among the early and fast developing branches of 20th century graph theory.
		Thus, random graphs are now popular characters in research papers in combinatorics, computer science, probability, mathematical physics, and biology, among others.
	asz.informatik.hu-berlin.de /en/school06 (566 words)

	Math 412, Sections X13 and X14 - Class Log
		Extremal problems on graphs: minimum number of edges in connected graphs on n vertices; minimum degree providing connectedness in simple graphs; large bipartite subgraphs.
		Petersen's theorem on 1-factors in 3-regular graphs, k-factors of graphs, Petersen's Theorem on 2-factors in 2k-regular graphs.
		A criterion for a graph to be k-connected.
	www.math.uiuc.edu /~kostochk/math312/classlog.html (363 words)

	Citations: Journal of Graph Theory - Turan, of (ResearchIndex)
		Turan: A note of welcome, Journal of Graph Theory 1 (1977), 7--9.
		According to a famous conjecture of Zarankiewicz [10] the crossing number of the complete graph K n,m with n and m vertices in its classes....
		In the next section, we give a completely di erent representation of planar graphs (see Theorem 2.3) If G is not planar then it cannot be drawn in the plane without crossing.
	citeseer.ist.psu.edu /context/95683/0 (621 words)

	Pál Turán - Wikipedia, the free encyclopedia
		He proved one of the first major results in extremal graph theory.
		Turán was sent to labour camps at various times from 1940 to 1944.
		After the war he was married twice and had 3 sons.
	en.wikipedia.org /wiki/Paul_Tur%C3%A1n (141 words)

	CS 572 \| Graduate Education \| Computer Science \| UIUC
		Distance and connectivity, matching and factors, vertex and edge colorings, perfect and imperfect graphs, intersection classes and intersection parameters, Turan's theorem, graph Ramsey theory, graph decomposition and other extremal problems.
		The perfect graph theorem, partitionable and imperfect graphs, special classes of perfect graphs (chordal, interval, etc.), intersection classes.
		Turan's theorem, graph Ramsey theory, graph decomposition, representation parameters.
	www.cs.uiuc.edu /graduate/courses.php?course=cs572 (104 words)

	Citations: An extremal problem in graph theory - Turan (ResearchIndex)
		The Maximum Number of Edges in a Graph of Bounded..
		Theorem 2.2 (Tur an s theorem) For positive integers p and t with p t 1, the maximum number of edges in a graph G on p vertices which does not contain a complete subgraph of size t 1 is T(p; t) Furthermore equality is....
		Tur'an, On an extremal problem in graph theory (in Hungarian), Matematikai 'es Fizikai Lapok 48 (1941), 436--452.
	citeseer.ist.psu.edu /context/1971006/0 (390 words)

	Gyorgy Turan's Publications
		Turan: The complexity of defining a relation on a finite graph, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 33 (1987), 277-288.
		Turan: On the definability of properties of finite graphs, Discrete Mathematics 49 (1984), 291-302.
		Turan: On the complexity of graph grammars, Acta Cybernetica 6 (1983), 271-280.
	www.math.uic.edu /~gyt/publications.html (1276 words)

	BackgroundMaterial (Site not responding. Last check: 2007-10-15)
		Theorem [Turan] If G is a graph on n vertices and G does not contain a complete subgraph of size k + 1, then G has at most t (n, k) edges.
		Then consider the graph H obtained from G by removing all edges between pairs of vertices in Y and making each vertex in Y adjacent to all vertices adjacent to x_1.
		This graph must have exactly the same degree sequence as G and thus is just G. Now the result follows easily by induction.
	www.math.gatech.edu /~trotter/Section7-Extremal.htm (551 words)

	[No title]
		If the graph is edge weighted, then the function returns a matching with maximum total weight." BipartiteMatchingAndCover::usage = "BipartiteMatchingAndCover[g] takes a bipartite graph g and returns a matching with maximum weight along with the dual vertex cover.
		The Petersen graph is identical to the generalized Petersen graph with n = 5 and k = 2." GetEdgeLabels::usage = "GetEdgeLabels[g] returns the list of labels of the edges of g.
		The third item in a Graph object is opts, a sequence of zero or more global options that apply to all vertices or all edges or to the graph as a whole.
	www.cs.uiowa.edu /~sriram/Combinatorica/NewCombinatorica.m (6443 words)

	Complete subgraphs and Turán's theorem
		We've seen the maximum size of a graph containing no path of a certain length.
		-partite graph is a graph with a vertex partition
		-partite graph, where any pair of vertices in different classes are joined.
	john.fremlin.de /schoolwork/graph/graph-theory/node8.html (161 words)

	Fields Institute - Programs-Thematic -Graph Theory & Optimization
		Extremal graph theory and extremal hypergraph theory are core areas of combinatorics.
		Extremal graph theory was basically started by Turán and was influenced greatly by the results of Paul Erdös: in fact, it was among his favorite fields.
		Both areas, especially the theory of hypergraphs, have developed considerably since the publication of Bollobás' excellent monograph "Extremal graph theory" (1978).
	www.fields.utoronto.ca /programs/scientific/99-00/graph_theory/mini-symposia/extremal_graph_theory (130 words)

	Publications of Erdos in ...
		Zbl 529.05027 • Erdös, Paul; Simonovits, Miklos, Supersaturated graphs and hypergraphs.
		Zbl 529.05053 • Erdös, Paul; Palmer, Edgar M.; Robinson, Robert W. Local connectivity of a random graph.
		Zbl 535.05049 • Erdös, Paul; Palka, Z. Trees in random graphs.
	www.zblmath.fiz-karlsruhe.de /MATH/general/general/erdos/1983.htm (473 words)

	Owen D. Byer (Site not responding. Last check: 2007-10-15)
		In particular, we fix a class $/cal[G]$ of graphs, and attempt to maximize and/or minimize various functions (graph invariants) from $/cal[G]$ to the nonnegative integers.
		We first consider the problem of finding (bounds for) f(v,e, λ), which is the maximum number of proper vertex colorings of a (v, e)-graph (a graph with v vertices and e edges) in λ colors.
		Finally, we show that the number of 4-cliques appearing in a 4-partite graph G of size e is at most $[e/sp2/over36]$ We conjecture that a K
	www.math.udel.edu /research/diss_abs/byer_abst.html (277 words)

	Comment on A Turan type problem... (Site not responding. Last check: 2007-10-15)
		These comments are also available in postscript of pdf format.
		Theorem 1.1 is sharp in the sense that for p \geq 4, t_p(n,K_k) is not obtained by the Turán graph.
		This can already be seen by the fact that the complete bipartite graph G=K_{\lfloor n/2-1 \rfloor, \lceil n/2+1 \rceil} has e_4(G) > e_4(T(n,3)).
	www.univie.ac.at /EMIS/journals/EJC/Volume_7/Comments/v7i1r47comments.html (177 words)

	Definition of Turan graph
		The Turan graph T(n,r) is the complete r-partite graph with n vertices whose partite sets differ in size at most 1.
		Clearly, the Turan graph T(n,r) does not contain a clique of size r+1.
		This is the best possible (with respect to the number of edges) among all graphs with n vertices (see Turan's theorem).
	www.wordiq.com /definition/Turan_graph (197 words)

	Read This: Proofs from THE BOOK
		The chapters are (7) Hilbert's third problem: decomposing polyhedra, (8) Lines in the plane and decompositions of graphs, (9) The slope problem, (10) Three applications of Euler's formula, (11) Cauchy's rigidity theorem, (12) The problem of the thirteen spheres, (13) Touching simplices, (14) Every large point set has an obtuse angle, and (15) Borsuk's conjecture.
		Some highlights in this section include an introduction to graph theory in chapter (8), an explanation of some spherical geometry and combined use of graph theory in chapter (12).
		Quite a bit of graph theory is used prior to this section of the book.
	www.maa.org /reviews/thebook.html (1314 words)

	Graph Theory Page (Site not responding. Last check: 2007-10-15)
		Chromatic number, graph colorings including Thomassen's proof that all planar graphs are 5 list colorable.
		(i) Introducing the probabilistic method in order to prove Erdos' famous theorem that there exist graphs with arbitrarily large chromatic number and girth (this demonstates the fact that the chromatic number can be greatly affected by "global" rather than "local" behavior).
		The latter has as a corollary a beautiful theorem in extremal graph theory which tells what edge density in large graphs is required in order to be guaranteed that one contains a fixed given subgraph H (the amazing formula is (f(H)-2)/(f(H)-1) where f(H) is the chromatic number of H).
	www.math.chalmers.se /~steif/graph.html (271 words)

	Comment on A Turan type problem...
		These comments are also available in postscript of pdf format.
		Theorem 1.1 is sharp in the sense that for p \geq 4, t_p(n,K_k) is not obtained by the Turán graph.
		This can already be seen by the fact that the complete bipartite graph G=K_{\lfloor n/2-1 \rfloor, \lceil n/2+1 \rceil} has e_4(G) > e_4(T(n,3)).
	www.combinatorics.org /Volume_7/Comments/v7i1r47comments.html (177 words)

	Problems in Topological Graph Theory
		Graphs that quadrangulate both the torus and Klein bottle
		Orientable genus of graphs of bounded nonorientable genus
		Geometric graphs with each edge crossing at most three others
	www.emba.uvm.edu /~archdeac/problems/problems.html (283 words)

	References for Turan (Site not responding. Last check: 2007-10-15)
		P Bundschuh, Review of Paul Turan's collected papers, Jahresberichte der Deutschen Mathematiker vereinigung 94 (1992), B3-5.
		M Simonovits, On Paul Turán's influence on graph theory, J.
		The graph theory bibliography of Paul Turán, J.
	www-groups.dcs.st-and.ac.uk /~history/References/Turan.html (246 words)

	[Lowerbounds, Upperbounds] » 2006 » February
		In an instance of the MDMST problem, we are given a graph G=(V,E) and a non-negative cost function c on edges, and the objective is to find a minimum cost spanning tree T under the cost function c such that maximum degree of T is minimized.
		Given two graphs $G$ and $H$, an $H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a graph isomorphic to $H$.
		For a non-empty graph $H$ we denote by gcd(H) the greatest common divisor of the degrees of $H$.
	magic.aladdin.cs.cmu.edu /2006/02 (3402 words)

	Graph Theory II - Math 532 (Site not responding. Last check: 2007-10-15)
		Text: Graph Theory by Ron Gould supplied off the web with additional materials provided.
		Also, show the Petersen graph has a nowhere zero 5-flow.
		Let X(G) be the number of edges of G not in triangles.
	www.mathcs.emory.edu /~rg/m532.html (183 words)

	Utility Functions used in Vega 0.4: Functions in EXMPLS20.M (Site not responding. Last check: 2007-10-15)
		CirculantGraph[n,l] constructs a circulant graph on n vertices, meaning the ith vertex is adjacent to the (i+j)th and (i-j)th vertex, for each j in list l.
		CirculantMatrix[n,l] constructs a circulant matrix on n vertices, meaning the ith vertex is adjacent to the (i+j)th and (i-j)th vertex, for each j in list l.
		Turan[n,p] constructs the Turan graph, the extremal graph on n vertices which does not contain K[p].
	vega.ijp.si /Htmldoc/USAGES/EXMPLS20.HTM (186 words)

	Problems in Topological Graph Theory
		Four-coloring all but three vertices of a toroidal graph
		Turan's brickyard problem: the crossing number of K(n,m)
		The expected number of regions in a random embedding of the complete graph
	www.emba.uvm.edu /~archdeac/newlist/problems.html (183 words)

	List of Publications
		Preliminary version appeared as Computation of the Ramsey Multiplicity of K_4 in the Proceedings of the Workshop on Computational Graph Theory and Combinatorics, Victoria, British Columbia (1999) 28-30.
		Proceedings of the 18-th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, 59 (1987) 155-164.
		Proceedings of the 17-th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, 55 (1986) 235-244.
	www.cs.rit.edu /~spr/PUBL/publ.html (1094 words)

	Publications of Paul Erdos (Site not responding. Last check: 2007-10-15)
		be the number of edges in Turán's graph, i.e.
		The following refinement of Turán's theorem is proved: if the graph
		is the number of edges in the graph induced by the neighbors of
	www.zblmath.fiz-karlsruhe.de /MATH/general/general/erdos/cit/51805044.htm (88 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		Class05: Graph Theory - Graph Colouring, Brook's Theorem
		Class10: Application of Turan's Theorem to parallel computation
		Class12: Graph Theory - SDR's and Konigs theorem
	www.math.cmu.edu /~af1p/Combinatorics/2003/coursenotes.html (58 words)

	Turán Graph (Site not responding. Last check: 2007-10-15)
		In other words, the Turán graph has the maximum possible number of
		Turán's Theorem gives the maximum number of edges
		so the Turán graph is given by the
	www.math.sdu.edu.cn /mathency/math/t/t426.htm (28 words)

	Working Papers
		The graphs for which all strong orientations are hamiltonian
		Rosenfeld, J. Zaks (Eds.): Convexity and Graph Theory.
		Graphs whose cycles are of length and modulus k
	www.or.uni-bonn.de /publications.de.html (13927 words)

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