Kempe chain

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.

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.

[4] In this case, Kempe chains are used to prove the idea that no vertex of degree four has to be touching four distinct colours different from itself.

We can set the colours as (in clockwise order) red, yellow, blue, and green.

In this situation, there can be a Kempe chain joining the red and blue neighbours or a Kempe chain joining the green and yellow neighbours, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be coloured with both red or blue and with green or yellow at the same time.