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| '''Extremal graph theory''' is a branch of the [[mathematics|mathematical]] field of [[graph theory]]. Extremal graph theory studies extremal (maximal or minimal) [[Graph (mathematics)|graphs]] which satisfy a certain property. Extremality can be taken with respect to different [[graph invariant]]s, such as order, size or girth. More abstractly, it studies how global properties of a graph influence local substructures of the graph.<ref>{{harvnb|Diestel|2005}}</ref>
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| For example, a simple extremal graph theory question is "which [[Forest (graph theory)|acyclic graph]]s on ''n'' vertices have the maximum number of edges?" The extremal graphs for this question are [[tree (graph theory)|trees]] on ''n'' vertices, which have ''n'' − 1 edges.<ref>{{Harvnb|Bollobás|2004|p=9}}</ref> More generally, a typical question is the following. Given a [[graph property]] ''P'', an invariant ''u'', and a set of graphs ''H'', we wish to find the minimum value of ''m'' such that every graph in ''H'' which has ''u'' larger than ''m'' possess property ''P''. In the example above, ''H'' was the set of ''n''-vertex graphs, ''P'' was the property of being cyclic, and ''u'' was the number of edges in the graph. Thus every graph on ''n'' vertices with more than ''n'' − 1 edges must contain a cycle.
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| Several foundational results in extremal graph theory are questions of the above-mentioned form. For instance, the question of how many edges can an ''n''-vertex graph have before it must contain as subgraph a [[clique (graph theory)|clique]] of size ''k'' is answered by [[Turán's theorem]]. Instead of cliques, if the same question is asked for complete multi-partite graphs, the answer is given by the [[Erdős–Stone theorem]].
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| ==History==
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| {{Quote box|quote=Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians.|source={{Harvtxt|Bollobás|2004}}|width=300px}}
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| Extremal graph theory started in 1941 when Turán proved [[Turán's theorem|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).<ref name="b104">{{Harvnb|Bollobás|1998|p=104}}</ref> Another crucial year for the subject was 1975 when Szemerédi proved [[Szemerédi's theorem|his result]] a vital tool in attacking extremal problems.<ref name="b104" />
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| ==Density results==
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| A typical result in extremal graph theory is [[Turán's theorem]]. It answers the following question. What is the maximum possible number of edges in an undirected graph ''G'' with ''n'' vertices which does not contain ''K''<sub>''3''</sub> (three vertices ''A'', ''B'', ''C'' with edges ''AB'', ''AC'', ''BC''; i.e. a triangle) as a subgraph? The [[complete bipartite graph]] where the partite sets differ in their size by at most 1, is the only extremal graph with this property. It contains
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| :<math> \left\lfloor \frac{n^2}{4} \right\rfloor</math> | |
| edges. Similar questions have been studied with various other subgraphs ''H'' instead of ''K''<sub>''3''</sub>; for instance, the [[Zarankiewicz problem]] concerns the largest graph that does not contain a fixed [[complete bipartite graph]] as a subgraph. [[Turán]] also found the (unique) largest graph not containing ''K''<sub>''k''</sub> which is named after him, namely [[Turán graph]]. This graph is the [[complete join]] of "k-1" independent sets (as equi-sized as possible) and has at most
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| :<math> \left\lfloor \frac{(k-2) n^2}{2(k-1)} \right\rfloor = \left\lfloor \left( 1- \frac{1}{k-1} \right) \frac{n^2}{2} \right\rfloor</math> | |
| edges. For ''C''<sub>''4''</sub>, the largest graph on ''n'' vertices not containing ''C''<sub>''4''</sub> has
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| :<math> \left(\frac{1}{2}+o(1)\right) n^{3/2}</math> | |
| edges.
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| ==Minimum degree conditions==
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| The preceding theorems give conditions for a small object to appear within a (perhaps) very large graph. At the opposite extreme, one might search for conditions which force the existence of a structure which covers every vertex. But it is possible for a graph with
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| :<math> \binom{n-1}{2} </math>
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| edges to have an isolated vertex - even though almost every possible edge is present in the graph - which means that even a graph with very high density may have no interesting structure covering every vertex. Simple edge counting conditions, which give no indication as to how the edges in the graph are distributed, thus often tend to give uninteresting results for very large structures. Instead, we introduce the concept of minimum degree. The minimum degree of a graph ''G'' is defined to be
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| :<math> \delta(G)=\min_{v\in G} d(v) . </math>
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| Specifying a large minimum degree removes the objection that there may be a few 'pathological' vertices; if the minimum degree of a graph ''G'' is 1, for example, then there can be no isolated vertices (even though ''G'' may have very few edges).
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| A classic result is [[Dirac's theorem on Hamiltonian cycles|Dirac's theorem]], which states that every graph ''G'' with ''n'' vertices and minimum degree at least ''n/2'' contains a [[Hamilton cycle]].
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| ==See also==
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| *[[Ramsey theory]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{Citation | last1=Bollobás | first1=Béla | author1-link=Béla Bollobás | title=Extremal Graph Theory | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-43596-1 | year=2004}}.
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| * {{Citation | last1=Bollobás | first1=Béla | author1-link=Béla Bollobás | title=Modern Graph Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-98491-9 | year=1998 | pages=103–144}}.
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| * {{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://diestel-graph-theory.com/index.html/ | publisher=Springer-Verlag | location=Berlin, New York | edition=4th | isbn=978-3-642-14278-9 | year=2010 | pages=169–198}}.
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| * M. Simonovits, Slides from the Chorin summer school lectures, 2006. [http://www.renyi.hu/~miki/BerlinG.pdf]
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| [[Category:Extremal graph theory| ]]
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