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| In [[extremal graph theory]], the '''forbidden subgraph problem''' is the following problem: given a graph ''G'', find the maximal number of edges in an ''n''-vertex graph which does not have a [[Glossary of graph theory#Subgraphs|subgraph]] [[graph isomorphism|isomorphic]] to ''G''. In this context, ''G'' is called a '''forbidden subgraph'''. <ref name=bollobas>''Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Probabilistic Combinatorics'', [[Béla Bollobás]], 1986, ISBN 0-521-33703-8, [http://books.google.com/books?id=LUUrTJ1Cx_0C&pg=PA53&lpg=PA53&dq=%22forbidden+subgraph+problem%22&source=bl&ots=o_ZcOV93Ep&sig=70mXvYYMHyDSGdmOIwpHMLpx86E&hl=en&ei=n3KjSfHMCInOsAOCttieAg&sa=X&oi=book_result&resnum=7&ct=result#PPA54,M1 p. 53, 54]</ref>
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| It is also called the '''Turán-type problem''' and the corresponding number is called the '''Turán number for graph''' ''G''. It is called so in memory of [[Pál Turán]], who determined this number for all ''n'' and all [[complete graph]]s <math>K_r,\, n\geq r \geq 3</math>. <ref>[http://books.google.com/books?id=4hWxnsLIfVAC&pg=PA254&dq=%22Turan-type+problem%22&lr= p. 254]</ref>
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| An equivalent problem is how many edges in an ''n''-vertex graph guarantee that it has a subgraph isomorphic to ''G''?<ref>"Modern Graph Theory", by Béla Bollobás, 1998, ISBN 0-387-98488-7, [http://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA123&dq=%22forbidden+subgraph+problem%22&lr=#PPA103,M1 p. 103]</ref>
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| The problem may be generalized for a set of forbidden subgraphs ''S'': find the maximal number of edges in an ''n''-vertex graph which does not have a subgraph isomorphic to any graph fom ''S''.<ref>Handbook of Discrete and Combinatorial Mathematics By Kenneth H. Rosen, John G. Michaels [http://books.google.com/books?id=C3tJsmWRQvkC&pg=PA590&dq=%22turan+number%22#PPA590,M1 p. 590]</ref>
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| ==See also==
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| *[[Turán number]]
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| *[[Subgraph isomorphism problem]]
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| *[[Forbidden graph characterization]]
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| *[[Zarankiewicz problem]]
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| ==References==
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| {{reflist}}
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| {{combin-stub}}
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| [[Category:Extremal graph theory]]
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Latest revision as of 10:39, 8 December 2014
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