Kempe chain: Difference between revisions
No edit summary |
clean up using AWB (7513) |
||
| Line 1: | Line 1: | ||
{{ |
{{Refimprove|date=December 2009}} |
||
In [[mathematics]], a '''Kempe chain''' is a device used mainly in the study of the [[four colour theorem]]. |
In [[mathematics]], a '''Kempe chain''' is a device used mainly in the study of the [[four colour theorem]]. |
||
| Line 21: | Line 21: | ||
==In terms of maps== |
==In terms of maps== |
||
{{Expand |
{{Expand section|date=June 2008}} |
||
<!-- here we need a pretty picture and a down-to-earth explanation NOT in terms of graph theory terminology --> |
<!-- here we need a pretty picture and a down-to-earth explanation NOT in terms of graph theory terminology --> |
||
==Application to the four colour theorem== |
==Application to the four colour theorem== |
||
{{Expand |
{{Expand section|date=June 2008}} |
||
<!-- 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 --> |
<!-- 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 --> |
||
==Other Applications== |
==Other Applications== |
||
Kempe-chains have been used to prove several theorems in ''precoloring extension''<ref>Michael O. Albertson: ''You Can't Paint Yourself into a Corner'', 189-194. J. Combin. Theory Ser. B 73, 1998.</ref> |
Kempe-chains have been used to prove several theorems in ''precoloring 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>. |
||
==References== |
==References== |
||
Revision as of 12:19, 27 December 2010
 | This article needs additional citations for verification. (December 2009) |
In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem.
History
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 incorrect, 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-colour theorem, a weaker form of the four-colour theorem.
Formal definition
The term "Kempe chain" is used in two different but related ways.
Suppose G is a graph with vertex set V, and we are given a colouring function
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.
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.
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
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.
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 paths, which is simpler than in the case of the first definition.
In terms of maps
![[icon]](/enwiki/wiki/File:Wiki_letter_w_cropped.svg){kind=small} | This section needs expansion. You can help by adding to it. (June 2008) |
Application to the four colour theorem
| This section needs expansion. You can help by adding to it. (June 2008) |
Other Applications
Kempe-chains have been used to prove several theorems in precoloring extension[1][2].