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.

翻译：我们证明每个不同于棱柱的2-球面图可用顶点4-着色，使得所有Kempe链均为森林。这蕴含以下三树定理：离散2-球面的树荫度为3。此外，可选取这三棵树使得每棵树均经过每个三角形。其推论是Bondy与Murty著作中基于A. Frank、A. Gyarfas和C. Nash-Williams工作的习题结论：平面图的树荫度小于等于3。