The four color theorem means you can color the United States in four colors. But can you color it in three? Try it before you read on.
The answer is no. Consider Nevada. It takes three colors to color that states that surround Nevada (Oregon, Idaho, Utah, Arizona and California) and then one more for Nevada.
I use this as an example of a heuristic in the P v NP book I'm working on. But what about the converse: Can a map have no internal states with an odd number of neighbors and still require four colors. I can come up with some examples but they all require either a lake separating states (like Lake Michigan) or more than three states coming together at a point (like the Four Corners of Utah, Arizona, New Mexico and Colorado).
I wrote this question in terms of planar graphs and asked it on Theory Q&A. Turns out there are no other examples. Take any map, where all internal states have an even number of neighbors with no lakes and no point that borders more than three states and that map can be three colored. Cool.
I'd like to find natural examples, real maps of anywhere where
1) At least two internal regions (to make it interesting)
2) All internal regions have an even number of neighbors
3a) The map is not three colorable (has lakes or more than three regions meeting in a point), or
3b) There are no lakes or more than three regions meeting in a point (so three colorable).
If you are the first to give me an example I use in the book, I'll send you a free copy of the book when it is published.