Kempe chain

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

$c:V\to S$

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

$c:E\to S.$

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.