Ultrametricity Transition in the Graph Colouring Problem

Abstract

We study numerically the (planar) graph colouring problem with q colours. For q = 4, when a perfect colouring can be achieved, the solutions are scattered “randomly” (as far as triangle correlations are concerned) in configuration space. On the contrary, for q = 3, colouring is always imperfect, but the optimal solutions seem to organize themselves in an ultrametric way. This could illustrate rather well the role of frustration on the configuration space landscape. We discuss the importance of the distance chosen.