Coloring a PlayDome (mathematics first)

I wanted to color my PlayDome so that adjacent tires had different colors. How many different colors of paint did I need to buy? By looking at the 5 large tires that ring the topmost tire, it is clear that I would need at least 3 different colors. But then the topmost tire will require a fourth color so that it has a color distinct from the other three used to color the 5 adjacent tires. So, I need to buy at least four different colors of paint. Do I need more, 5 colors?

A famous theorem from graph theory, the Four Color Theorem, allows us to conclude that we should be able to color the PlayDome using only four colors. By making the tires the vertices of a graph, and then adding edges where bolts go, we obtain a planar graph on 21 vertices with 50 edges. Since this graph is planar, the Four Color Theorem asserts that we will not require more than 4 colors.

Now, how to achieve such a coloring? I started with a greedy algorithm - order the colors, and use the first possible color that will "work." However, this leads to a lot of tires using the first color, and not too many using the last color. What is probably desired is what I might call an "equitable" coloring - about the same number of tires of each color. In this case, that would be 3 sets of 5 tires and one set of 6 tires, where each set uses the same color.

So to achieve an equitable coloring, I did some Kempe interchanges. Now, if none of this makes much sense, do not despair - this is good fodder for an undergraduate honors thesis in mathematics (and it has been!).

Coloring Plan

So, if you have lasted this long, here's how I colored my dome: Start with Curt's detailed schematic (page 2 of the plans) and choose any one of the buried tires to be tire number 1. Work your way around the periphery, alternating buried and non-buried large tires, finally marking tire number 10. Now continue clockwise to the small tire that will end up above tire number 1, and call this tire number 11. Continue clockwise, alternating small and large tires until you count up to tire 20. Finally, call the central, capstone tire, number 21.

Now color as follows:

You can of course substitute or rearrange the colors to suit your preferences.

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Created: June 25, 1996, Updated: June 25, 1996.