  We prove that every 2-sphere graph different from a prism can be vertex
4-colored in such a way that all Kempe chains are forests. This implies the
following three tree theorem: the arboricity of a discrete 2-sphere is 3.
Moreover, the three trees can be chosen so that each hits every triangle. A
consequence is a result of an exercise in the book of Bondy and Murty based on
work of A. Frank, A. Gyarfas and C. Nash-Williams: the arboricity of a planar
graph is less or equal than 3.
