In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem. Intuitively, it is a connected chain of vertices on a graph with alternating colours.

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 success of valid modern proofs, such as the first successful one by Kenneth Appel and Wolfgang Haken. Furthermore, the method is used in the proof of the five color theorem by Percy John Heawood, a weaker but more easily proven version of the four colour theorem.

The term "Kempe chain" is used in two different but related ways.

Suppose G is a graph with vertex set V, with a given 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. With these additional conditions, a and b are two out of the four colours available, and every element of the (a, b)-Kempe chain has neighbours in the chain of only the other colour.

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.