# How to colour the US map with Yellow, Green, Red and Blue to minimize the number of states with the color of Green

I want to colour the US (only the states) map with Yellow, Green, Red and Blue. I was wondering what would be the lowest number of states with the colour of Green. We can of course use the other colours as much as we want. Please note that I want to follow the Four Color Theorem rules.

Motivation: I am studying graph theory and I want to know if there is a way that we could limit the use of the fourth colour as much as possible. This is not a homework problem.

My attempt: I have tried many variations and can limit it to 6 and it seems like the minimum possible but there are many possibilities to try ($$4^{50}$$). Therefore I was wondering if there is a simpler method? Thank you in advance.

Clarification: I am interested in only the mainland of USA. For states like Michigan that are split, I used the same colour for both parts.

• you would need to agree on a favorite version of the graph. In the actual US, there are islands, states split into disconnected regions, other things forbidden Apr 14 '19 at 3:36
• blog.computationalcomplexity.org/2006/05/… They correctly point out that three colors cannot work, as Nevada has an odd number of neighbors Apr 14 '19 at 3:40
• thank you for your suggestion, I made a few clarifications. Apr 14 '19 at 4:00
• @WillJagy Maine has an odd number of neighbours too, but that's not much of a problem. Apr 14 '19 at 17:09
• There are not infinite possibilities. There is only $4^{50}$ ways of assigning colors to all the states even if we consider no other constraints or symmetries. Apr 14 '19 at 20:10

The minimum is two states that use the fourth color. Nevada and its five neighbors cannot be colored with only three colors, and similarly West Virginia and its five neighbors cannot be colored with only three colors. In both cases, once you color the center state one color (say, red), you can't use it again on its neighbors: without using green, they'd have to alternate yellow-blue-yellow-blue, but because the number of neighbors is odd, you'd get stuck at the end.

(In the comments, David K points out that Kentucky is a third state with the same problem: it has seven neighbors. But this doesn't force us to use a third green state, because Kentucky and West Virginia share a border and some common neighbors.)

Using only two green states is possible. If we color Arizona (dealing with the Nevada situation) and Ohio (dealing with West Virginia and Kentucky) both green, then the remainder of the map can be completed using only blue, red, and yellow:

Adjacencies between the states may be easier to see here.

• just curious -- did you write code to do this, or did you do this by hand? Apr 14 '19 at 4:56
• @antkam By hand. I found two subgraphs where a fourth color is forced, and chose a state from each of them to color green that seemed to be a good choice. Then I just tried to color the rest with three colors - and for that, once you color the first two states, most of the rest of the map is forced, except for a few states like Maine. Apr 14 '19 at 5:56
• @WillJagy NJ and MD don't touch because DE interposes. All the other edges exist. Apr 14 '19 at 7:35
• Kentucky has seven neighbors, but coloring Ohio green fixes that as well as WV. I also notice that you managed to use four colors for UT, CO, AZ, and NM, which makes the map a little clearer but I think is not strictly required by the usual rules, that is, under the usual rules (as far as I know) you could make AZ yellow and CA green. Apr 14 '19 at 13:08
• Point of scientific interest: points such as the Four Corners en.wikipedia.org/wiki/Four_Corners_Monument happen only when legislated by humans. In nature, existing edges, such as riverbeds, split on one side in an orthogonal manner, creating a T junction that is, well, typically fractal. One can do an experiment with a plate of wet clay, heat it under a lamp, and watch edges appear as the mud dries out. I will try to find out the name for this... Apr 14 '19 at 15:58