Is the World Map Four-Colorable?

Is it possible to color a map of the world’s countries using four colors such that no two countries that border each other have the same color?

If by this point in the post you have already reached the answer “Yes, use the Four-Color Theorem”,* hold your horses, because the Four-Color Theorem applies to a division of the plane into contiguous regions, and an assertion that countries are contiguous regions is extremely nontrivially untrue: tons of countries own islands, and many countries own exclaves, like the USA’s Alaska and Russia’s Kaliningrad Oblast, completely disconnected from the main part of the country.

(*And if at this point reading the paragraph you reached the objection “commas are supposed to go inside quotation marks”,* you should note that for one, only Americans do this, and second, the reason Americans do this is because back-in-the-day it was considered of aesthetic value to put tiny pieces of punctuation inside the quotation marks, due to how typography back then came out. In the modern world, this issue is nonexistent. Like with slavery and imperial units, this is another place where the United States has decided that the fact that we have been doing it is more important than any advantage in reason, so if you think that tradition itself is a dumb reason to back a decision, that is, if you think the world should have nice things, the choice to make is to place commas and periods outside quotation marks when it makes the punctuation of the sentence make logical sense.)

In fact, if you perform an image search for four-colored world maps, chances are you’ll be hit with a first page of exclusively incorrect maps.

google_image_search_four_color_map

yahoo_image_search_four_colors

duckduckgo_image_search_four_colors

Maps 1 through 5 color France and the Netherlands the same color, whereas these two countries do in fact border each other, that is, on a Caribbean island. On top of this, Map 2 is not an accurate map of the world: it shows Czechoslovakia undivided and Eritrea unseparated from Ethiopia, implying that the time in history the map corresponds to is far before Montenegro separated from Serbia, which the map also claims. Map 4 seems to be an attempt at four-coloring the world while also considering bodies of water as entities, given that the Caspian Sea is colored a different color from the World Ocean; it is quite a respectable attempt, but it once again fails the above criterion.

Map 6 correctly decides to color France and the Netherlands differently, but fails to color Turkey and Azerbaijan differently: the two countries share one of the shortest borders in the world because Azerbaijan includes the Naxçıvan exclave (which, incidentally, Map 2 and Map 3 also fail to separate). Maps 7, 8, and 9 also fail at at least the France-Netherlands situation.

With the aid of the map charting tools of mapchart.net, I drew up an actual four-color world map.

actual_four_color_map

(Notice that I chose colors for the UK, Argentina, and Chile such that Antarctica resolves correctly no matter which of them you listen to—if you decide to recognize claims in Antarctica as legitimate and favor the Argentine lines over the British lines, the Argentine claim is still separated from the Norwegian claim by 5 degrees of longitude.)

This shows that the world can be colored using four colors right?

…not necessarily. This is one of many possible conceptions of the political borders of the world. This map by itself doesn’t show that the map of the world is four-colorable, regardless of which country’s diplomatic recognitions you follow. In fact, this map of the world is actually the world in not a single UN-recognized country’s eyes, because every single country of the world recognizes the People’s Republic of China xor the Republic of China as being the one true China.

So can the map become un-four-colorable depending on how border disputes resolve?

Yes. Amboyna Cay is claimed by both Chinas, Vietnam, the Philippines, and Malaysia. That’s a five-country dispute, so they could in fact resolve the dispute by drawing borders so ridiculous that parts owned by all five countries border parts owned by all four others.

How about something far more realistic than that?

Not really. If the Nagorno-Karabakh Republic successfully becomes a country and then Armenia decides to cede a portion of it that borders Turkey to it, this would necessitate a fifth color (as Nagorno-Karabakh would then border Turkey, Armenia, Azerbaijan, and Iran, which all border each other), but this is still not remotely likely.

But there’s something important that this investigation of the fluidity of country borders should remind us of: the world map changes over time. In particular, was there a point in history at which the world map could not be colored by just four colors?

Surely back in the days when the UK, France, Spain, and Portugal had empires stretching across the globe, the world map was un-four-colorable at some point, right?

Indeed, this is the case. From -39 CL to -31 CL, every pair of {United Kingdom, France, Portugal, Belgium, Germany} bordered each other. On the European mainland, each pair among France, Belgium, and Germany shared a border. British Rhodesia bordered German East Africa, the Belgian Congo Free State, and Portuguese Angola. The British Gold Coast bordered French West Africa. The Congo Free State also bordered Portuguese Angola, French Guinea bordered Portuguese Guinea, and Portuguese Mozambique bordered German East Africa.

So indeed, even though the world map satisfies four-colorability right now, it didn’t always.

UPDATE: This post now has an addendum on the planarity of the world map graph.

Advertisements

One thought on “Is the World Map Four-Colorable?

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s