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| {{distinguish|Tutte homotopy theorem|Tutte's spring theorem}}
| | Hi, everybody! My name is Lettie. <br>It is a little about myself: I live in Norway, my city of Kristiansand S. <br>It's called often [http://Www.Channel4.com/news/Northern Northern] or cultural capital of NA. I've married 3 years ago.<br>I have two children - a son (Alexandria) and the daughter (Eddy). We all like Ice hockey.<br><br><br><br>[http://www.google.de/url?url=https://www.facebook.com/Hostgator1CentCoupon&rct=j&q=&esrc=s&sa=U&ei=GDopVY7fOYKAafPCgQg&ved=0CCgQFjAB&usg=AFQjCNFnaRHJ6rbzPmQRrBRNgnvI0HokiQ google.de]my homepage; [http://support.groupsite.com/entries/32046360-Straightforward-Options-To-Assist-You-To-Get-High-Quality-Internet-Hosting Hostgator Coupon Code] |
| In the [[mathematical]] discipline of [[graph theory]] the '''Tutte theorem''', named after [[William Thomas Tutte]], is a characterization of [[graph (mathematics)|graphs]] with [[perfect matching]]s. It is a generalization of the [[marriage theorem]] and is a special case of the [[Tutte–Berge formula]].
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| == Tutte's theorem ==
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| A graph, {{math|''G'' {{=}} (''V'', ''E'')}}, has a [[perfect matching]] [[if and only if]] for every subset {{math|''U''}} of {{math|''V''}}, the [[Glossary of graph theory#Subgraphs|subgraph]] induced by {{math|''V'' − ''U''}} has at most {{math|<nowiki>|</nowiki>''U''<nowiki>|</nowiki>}} [[connected component (graph theory)|connected component]]s with an odd number of [[vertex (graph theory)|vertices]].<ref>{{harvtxt |Lovász|Plummer|1986| p=84}}</ref>
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| ==Proof==
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| First we write the condition:
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| :<math>(*) \qquad \forall U \subseteq V, \quad o(G-U)\le|U|</math>
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| '''Necessity of (∗):''' Consider a graph {{math|''G''}}, with a perfect matching. Let {{math|''U''}} be an arbitrary subset of {{math|''V''}}. Delete {{math|''U''}}. Let {{math|''C''}} be an arbitrary odd component in {{math|''G'' − ''U''}}. Since {{math|''G''}} had a perfect matching, at least one vertex in {{math|''C''}} must be matched to a vertex in {{math|''U''}}. Hence, each odd component has at least one vertex matched with a vertex in {{math|''U''}}. Since each vertex in {{math|''U''}} can be in this relation with at most one connected component (because of it being matched at most once in a perfect matching), {{math|''o''(''G'' − ''U'') ≤ <nowiki>|</nowiki>''U''<nowiki>|</nowiki>}}.
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| '''Sufficiency of (∗):''' Let {{math|''G''}} be an arbitrary graph satisfying (∗). Consider the expansion of {{math|''G''}} to {{math|''G∗''}}, a maximally imperfect graph, in the sense that {{math|''G''}} is a spanning subgraph of {{math|''G''∗}}, but adding an edge to {{math|''G∗''}} will result in a perfect matching. We observe that {{math|''G∗''}} also satisfies condition (∗) since {{math|''G∗''}} is {{math|''G''}} with additional edges. Let {{math|''U''}} be the set of vertices of degree {{math|<nowiki>|</nowiki>''V''<nowiki>|</nowiki> − 1}}. By deleting {{math|''U''}}, we obtain a disjoint union of complete graphs (each component is a complete graph). A perfect matching may now be defined by matching each even component independently, and matching one vertex of an odd component {{math|''C''}} to a vertex in {{math|''U''}} and the remaining vertices in {{math|''C''}} amongst themselves (since an even number of them remain this is possible). The remaining vertices in {{math|''U''}} may be matched similarly, as {{math|''U''}} is complete.
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| This proof is not complete. Deleting {{math|''U''}} may yield a disjoint union of complete graphs, but the case where it does not is the more interesting and difficult part of the proof.
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| == See also == | |
| * [[Bipartite matching]]
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| * [[Hall's theorem]]
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| * [[Petersen's theorem]]
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| ==Notes== | |
| {{reflist}}
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| ==References==
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| * {{Cite book
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| | last=Bondy | first=J. A.
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| | last2=Murty | first2=U. S. R.
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| | title=Graph theory with applications | year=1976 | publisher=American Elsevier Pub. Co. | location=New York | isbn=0-444-19451-7}}
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| * {{Cite book
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| | last1=Lovász | first1=László | author1-link=László Lovász
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| | last2=Plummer | first2=M. D. | author2-link=Michael D. Plummer
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| | title=Matching theory | year=1986 | publisher=North-Holland | location=Amsterdam | isbn=0-444-87916-1}}
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| [[Category:Matching]]
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| [[Category:Theorems in graph theory]]
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| [[Category:Articles containing proofs]]
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Hi, everybody! My name is Lettie.
It is a little about myself: I live in Norway, my city of Kristiansand S.
It's called often Northern or cultural capital of NA. I've married 3 years ago.
I have two children - a son (Alexandria) and the daughter (Eddy). We all like Ice hockey.
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