Comment by emil-lp

> The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.

Well, the current conjecture is that it hasn't to do with coloring at all, but how many, on a map (partitioning of plane into connected regions), regions can pairwise touch: 4.

(That is: If someone proves Hadwiger's conjecture, then the four colour theorem follows.)