We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most
ϵ(Δ)=MΔ26 colours, for a fixed constant M. This implies
that X (and the associated median graph) isometrically embeds in the
Cartesian product of at most ϵ(Δ) trees, and that the event
structure whose domain is X admits a nice labeling with
ϵ(Δ) labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire