Are there known **upper** and **lower** bounds for the number of colors required for drawing a multi-dimensional tesselation described by a Coxter-Dynkin diagram?

By skimming through “**Higher-Dimensional Analogues of the Map Coloring Problem**” by Bhaskar Bagchi and Basudeb Datta I assume that the upper-bound number of colors given the number of dimensions is **3^n+3^(n-1)-2**

Nonetheless, I’m not sure how the curvature of the space influences this.

Do you have any other suggestions or approaches?

How does the 4-color theorem generalize for spaces of higher-dimensions and non-negative curvature described by Coexter-Dynkin diagrams?