Scaled inverse chi-squared distribution: Difference between revisions

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{{distinguish|Tutte homotopy theorem|Tutte's spring theorem}}
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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]].
 
== Tutte's theorem ==
A graph, {{math|''G''&nbsp;{{=}}&nbsp;(''V'',&nbsp;''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''&nbsp;&minus;&nbsp;''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>
 
==Proof==
First we write the condition:
 
:<math>(*) \qquad \forall U \subseteq V, \quad o(G-U)\le|U|</math>
 
'''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''&nbsp;&minus;&nbsp;''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''&nbsp;&minus;&nbsp;''U'')&nbsp;≤&nbsp;<nowiki>|</nowiki>''U''<nowiki>|</nowiki>}}.
 
'''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>&nbsp;&minus;&nbsp;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.
 
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.
 
== See also ==
* [[Bipartite matching]]
* [[Hall's theorem]]
* [[Petersen's theorem]]
 
==Notes==
{{reflist}}
 
==References==
* {{Cite book
| last=Bondy  | first=J. A.
| last2=Murty | first2=U. S. R.
| title=Graph theory with applications | year=1976 | publisher=American Elsevier Pub. Co. | location=New York  | isbn=0-444-19451-7}}
* {{Cite book
| last1=Lovász    | first1=László | author1-link=László Lovász
| last2=Plummer | first2=M. D. | author2-link=Michael D. Plummer
| title=Matching theory | year=1986 | publisher=North-Holland | location=Amsterdam | isbn=0-444-87916-1}}
 
[[Category:Matching]]
[[Category:Theorems in graph theory]]
[[Category:Articles containing proofs]]

Revision as of 13:48, 23 February 2014

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