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| | I am Oscar and I totally dig that title. Hiring has been my occupation for some time but I've already applied for another 1. South Dakota is her beginning place but she needs to transfer because of her family. What I adore performing is taking part in baseball but I haven't produced a dime with it.<br><br>Here is my blog ... [http://Www.Asseryshit.com/groups/valuable-guidance-for-successfully-treating-yeast-infections/ home std test kit] |
| In [[mathematics]], a '''Kempe chain''' is a device used mainly in the study of the [[four colour theorem]].
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| ==History==
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| Kempe chains were first used by [[Alfred Kempe]] in his attempted proof of the four colour theorem. Even though his proof turned out to be incomplete, the method of Kempe chains is crucial to the successful modern proofs (Appel & Haken, Robertson et al., etc.). Furthermore, the method is used in the proof of the [[Five color theorem|five-colour theorem]], a weaker form of the four-colour theorem.
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| ==Formal definition==
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| The term "Kempe chain" is used in two different but related ways.
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| Suppose ''G'' is a [[Graph (mathematics)|graph]] with vertex set ''V'', and we are given a colouring function
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| : <math>c : V \to S</math>
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| where ''S'' is a finite set of colours, containing at least two distinct colours ''a'' and ''b''. If ''v'' is a vertex with colour ''a'', then the (''a'', ''b'')-Kempe chain of ''G'' containing ''v'' is the maximal connected subset of ''V'' which contains ''v'' and whose vertices are all coloured either ''a'' or ''b''.
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| The above definition is what Kempe worked with. Typically the set ''S'' has four elements (the four colours of the four colour theorem), and ''c'' is a [[proper colouring]], that is, each pair of adjacent vertices in ''V'' are assigned distinct colours.
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| A more general definition, which is used in the modern computer-based proofs of the four colour theorem, is the following. Suppose again that ''G'' is a graph, with edge set ''E'', and this time we have a colouring function
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| : <math>c : E \to S.</math> | |
| If ''e'' is an edge assigned colour ''a'', then the (''a'', ''b'')-Kempe chain of ''G'' containing ''e'' is the maximal connected subset of ''E'' which contains ''e'' and whose edges are all coloured either ''a'' or ''b''.
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| This second definition is typically applied where ''S'' has three elements, say ''a'', ''b'' and ''c'', and where ''V'' is a [[cubic graph]], that is, every vertex has three incident edges. If such a graph is properly coloured, then each vertex must have edges of three distinct colours, and Kempe chains end up being [[path (graph theory)|path]]s, which is simpler than in the case of the first definition.
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| ==In terms of maps==
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| {{Expand section|date=June 2008}}
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| <!-- here we need a pretty picture and a down-to-earth explanation NOT in terms of graph theory terminology -->
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| ==Application to the four colour theorem==
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| {{Expand section|date=June 2008}}
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| <!-- Here we need an explanation of e.g. how Kempe chains can be used to reduce a vertex of degree four in the 4ct case, and how to reduce a vertex of degree 5 in the 5ct case, again with some pretty pictures -->
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| ==Other Applications==
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| Kempe-chains have been used to solve problems in ''coloring extension.''.<ref>Michael O. Albertson: ''You Can't Paint Yourself into a Corner'', 189-194. J. Combin. Theory Ser. B 73, 1998.</ref><ref>Michael O. Albertson and Emily H. Moore: ''Extending graph colorings'', 83-95. J. Combin. Theory Ser. B 77, 1999.</ref>
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Kempe Chain}}
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| [[Category:Graph coloring]]
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I am Oscar and I totally dig that title. Hiring has been my occupation for some time but I've already applied for another 1. South Dakota is her beginning place but she needs to transfer because of her family. What I adore performing is taking part in baseball but I haven't produced a dime with it.
Here is my blog ... home std test kit