Four colour theorem on a sphere should be five colour theorem?

Question

I have read many times that the four colour theorem which applies to planes also applies to spheres.

But I have recently been thinking about this and wondered if it is really true.

In one dimensions, for a line, there is the two colour theorem, in that any one dimensional line can be partitioned using a minimum of two colours so that no two colours which are the same touch.

But for a circle, you need a minimum of three colours, since if the partition has an odd number of partitions, a two colouring will result in two colours which are the same touching one another at the point the line reconnects with itself.

It seems to me that the same principle should apply with a sphere as opposed to a plane. Where the sphere meets itself again, you may need an extra colour.

Hence the four colour theorem for the plane should be the five colour theorem for the sphere.

Is this correct? If not, why not?

Answer 1

The four-color theorem can be thought of as saying: any graph that embeds in $\mathbb R^2$ is 4-colorable. And similarly with the 2-color theorem on $\mathbb R$ and the 3-color theorem on $S^1.$ Since we can view $\mathbb R\subset S^1$, any graph that embeds in $\mathbb R$ also embeds in $S^1$. But as you said, the complete graph $K_3$ is homeomorphic to $S^1$, so this can’t be restricted to $\mathbb R.$ However, a graph embedded in $S^2$ always misses at least one point. Then the graph also embeds into $S^2-\text{pt}=\mathbb R^2.$ That way, we can see that embedding into the sphere or the plane are the same thing, in contrast to $S^1$ and $\mathbb R$.

Answer 2

One strategy for doing a 2-coloring of a circle is to make a cut in the circle. The circle can then be unrolled into a line, and we can do a 2-coloring of the line. Then glue the line back together into a circle.

The problem with this strategy is that when you cut the circle, you are also cutting a section of the circle. Whatever section you put the cut in, that section will be split into two sub-sections, and there's no guarantee that the 2-coloring on the line will give the same color to both sub-sections.

But when you cut a hole in a sphere, you don't have the same problem. As long as the hole is in the interior of a section, the section isn't being split into two sub-sections, and the graph equivalence isn't affected. For instance, imagine taking a map of Europe and trying to do a 4-coloring. If you take a pair of scissors and cut Paris out of the map, will that affect the coloring at all?

Of course, this then relies on the fact that a sphere with a hole cut out of it can be deformed to be embedded in a plane, but that's a pretty basic topological fact.