Ultrametricity, frustration and the graph colouring problem

Using techniques of numerical taxonomy the authors study ultrametricity in the planar graph colouring problem with three and four colours. Evidence of a non-trivial hierarchical organisation is given for the case q=3. For q=4 they find a simpler ultrametricity with only equilateral triangles of side length d approximately=0.77. This example shows that lack of frustration does not rule out ultrametricity completely although it appears in a rather trivial way.