On Tuesday, March 4, 2014 12:24:11 PM UTC-6, swtch...@gmail.com wrote: > ABABABA > > BABABAB > > ABABABA > > > > This checkered coloring scheme would not work since same colors are used in the adjacent countries in this case.
You seem to not understand the assumptions of the 4-color map theorem. In particular, "adjacency" only exists when countries share a common border of positive length; a single point (e.g., the four state corner in the US) does not count as adjacency between diagonally opposed countries.
If you ignore the hypotheses, it is usually easy to come up with a "counterexample". For instance, it is easy to draw a map on a torus that requires more than 4 colors; it does not invalidate the 4-color map theorem, however, because that is a theorem about *planar* maps, in which countries are *connected* and point adjacencies do not count.
Next time, perhaps you should familiarize yourself with the actual theorem you want to discuss, instead of addressing a straw man.