I studied and understood the proof of the 4 color theorem, which states that given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color; now, I encountered with the following statement
For any separation of a torus into contiguous regions, the regions can be colored using at most seven colors so that no two adjacent regions have the same color.
I have been looking for a proof of such statement; which I cannot find, I was hoping if anyone could give me some reference of where could I find such proof, are maybe give some hint to prove it myself. Thanks in advance
It can be shown that for any orientable surface, the minimum number of colors is given by $$ \left \lfloor \frac{7+\sqrt{1+48g}}{2} \right \rfloor $$ where $g$ is the genus of the surface (the "number of holes"). Since for a torus $g=1$, it follows that the minimum number of colors equals $7$.
Note that the formula does not hold for $g=0$, otherwise it would be an elegant and short proof of the $4$ color theorem in the plane.