The Mathematics of Map Coloring

– The Four-Color Theorem

Roger HouseOLLI Science Club

Sonoma State University2013 October 22

Copyright © 2013 Roger House

The Problem

You have a map of some sort, say a map of the counties of England, and you wish to color each county in such a way that no two adjacent counties are the same color.

Of course, you could use a different color for each county, but there's a condition:

Use as few colors as possible.

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Counties of England

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The Counties, Colored

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Why Counties of England?

Some time before 1852 October 23, Francis Guthrie (1831-99) was coloring a map of the counties of England when several questions occurred to him:

What is the minimum number of colors that would work for the counties of England?

What is the minimum number of colors that would work for any map?

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1852 October 23

How do we know Francis Guthrie came up with the problem before 1852 October 23?

Because on that date his brother, Frederick Guthrie (1833-86), sent a letter to Augustus de Morgan (1806-71) stating the problem and crediting Francis with it.

On the very same day, de Morgan sent a letter to Sir William Rowan Hamilton (1805-65) telling him of the problem.

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How to begin?

Mathematicians present their results in tidy, logical, step-by-step, pristine papers in learned journals.

They virtually never arrive at their results in the manner suggested by their papers.

They start with pencil and paper (or computers) and try this and try that, and offer sacrifices to the Math Mistress.

We will do the same.

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1 country – how many colors?

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1 country – 1 color

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2 countries – how many colors?

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2 countries – 2 colors

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3 countries – how many colors?

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3 countries – 3 colors

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3 countries – how many colors?

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3 countries – 2 colors

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4 countries – how many colors?

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4 countries – 2 colors

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4 countries – how many colors?

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4 countries – 3 colors

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4 countries – how many colors?

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4 countries – 4 colors

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Important fact

The previous example proves something rather important about map coloring:

The minimum number of colors required to color a map so that no two adjacent countries are the same color is four.

We now have a lower bound on the number we seek.

Can we find an upper bound?

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Do the math

Useful facts leading to an upper bound:

1. The Euler characteristic of a map is two.

2. Every map contains at least one country with five or fewer neighbors.

3. At most six colors are needed to color any map.

Note that this last result gives us an upper bound on the number of colors needed.

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Leonhard Euler (1707-83)

Leonhard Euler discovered just about everything:

Euler angles Euler totient function

Euler's formula Euler constant

Eulerian path Euler-Lagrange equation

Euler method Euler identity

His collected works: 76 volumes, so far...

We focus on the Euler characteristic.

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F + V – E = ?

2+4-4=?

3+6-7=?

4+8-10=?

5+10-13=?

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Euler Characteristic

F+V-E is called the Euler characteristic.

It certainly looks like it is always 2.

In fact, this is the case for maps drawn in the plane or on a sphere.

For a map drawn on a doughnut, things are different:

The Euler characteristic on a torus is zero.

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5 or fewer neighbors

Theorem: Every map has at least one country with 5 or fewer neighbors.

Proof: Assume otherwise, i.e., a map exists in which every country has 6 or more neighbors. A typical face:

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5 or fewer neighbors

The proof proceeds by using the Euler characteristic to show that the assumption that every country has 6 or more neighbors leads to a contractiction.

Thus there must be at least one country with 5 or fewer neighbors. █

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Six Colors Suffice

Theorem: Every map can be colored with no more than 6 colors in such a way that no two countries sharing a common boundary have the same color.

Proof: Assume otherwise, i.e., maps exist which require more than 6 colors.

Now use the fact just proved that a country with 5 or fewer neighbors exists to arrive at a contradiction. █

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Pick a number: 4, 5, or 6?

So now we know that the minimum number of colors required to color any map so that no two adjacent countries have the same color is 4, 5, or 6.

Well, which is it? 4? 5? or 6?

It looks as if 4 is the answer, but is it?

For a mathematician, this is an unsatisfactory situation.

“Wir müssen wissen!”

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No activity for several decades

Except for a couple of letters and a book review, there is no mention of the Four Color Problem from 1852 to 1878.

On 1878 July 13, Arthur Cayley (1821-95) inquired of the mathematical section of the Royal Society if a solution had been submitted.

None had, but this focused attention on the problem.

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Solved, at last

In 1878 Sir Alfred Bray Kempe (1849-1922) announced in Nature that he had found “the solution of a problem which recently achieved some renown.”

In 1879 his proof was published in the American Journal of Mathematics Pure and Applied.

Problem solved: Four colors suffice.

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How did he do it?

Kempe developed a very clever approach using chains of alternating colors.

These chains have since come to be known as Kempe chains.

Alas, we don't have sufficient time to go into the details of his proof, not even a sketch of the proof.

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So that's it for map coloring

Kempe's proof settled things – after 1879 not much was heard about map coloring.

Until …

In 1890, eleven years after Kempe's proof came out, Percy John Heawood (1861-1955) published a paper entitled Map-colour theorem.

The paper contained this map:

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Trouble in paradise

Heawood applied Kempe's chain technique to this map of 25 countries.

In fact, he applied the technique twice, using two chains simultaneously.

Kempe had not noticed that applying his chains twice simultaneously would not work in all cases.

In short, Heawood showed that Kempe's 'proof' was, in fact, not a proof.

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Ouch!

Heawood's example demolished Kempe's proof, i.e., Kempe's proof had a hole in it.

Kempe had not realized that applying his chains twice simultaneously could lead to an improper coloring.

So, suddenly, the Four Color Theorem once again became the Four Color Conjecture.

However, all is not lost ...

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The Five Color Theorem

Heawood showed that a modification of Kempe's method resulted in a proof of the Five Color Theorem.

So now we know that the minimum number of colors needed is 4 or 5.

What is needed to decide if 4 colors suffice?

A proof that works for all maps, i.e., a proof that works for an infinite number of maps.

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Maybe 4 colors aren't enough?

What is needed to decide if 5 colors suffice?

Just one map!

And, of course, an accompanying argument showing that the map requires 5 colors.

On 1975 April 1, Martin Gardner, editor for many years of the Mathematical Games column in Scientific American, published this map:

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Gardner: 5 colors are needed

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80+ years had gone by ...

What happened between 1890 and 1975?

Lots.

Many mathematicians tried their hands at proving the Four Color Conjecture.

But none succeeded in finding a proof.

Is that because 5 colors are needed?

Is Gardner's map a counterexample?

Note again the date on his map: April 1.

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Garner's map: 4 colors work

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At last!

But a solution was near at hand.

In 1976, Kenneth Appel (1932-) and Wolfgang Haken (1928-), two mathematicians at the University of Illinois at Urbana-Champaign, announced a proof.

In 1977 in the Illinois Journal of Mathematics they published their proof: "Every Planar Map is Four Colorable”.

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Some reactions

“I am willing to accept the Appel-Haken proof – beggars cannot be choosers.”

“To most mathematicians, however, the proof of the four-color conjecture is deeply unsatisfactory.”

“God wouldn't let the theorem be proved by a method as terrible as that!”

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What's the problem?

A key part of the proof was a computer program.

It analyzed 1476 cases, requiring 1200 hours of machine time.

Mathematicians were used to reading arguments presented in papers.

How could they verify that the proof was correct?

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Eventually: Acceptance

“To reject the use of computers as what one may call 'computational amplifiers' would be akin to an astronomer refusing to admit discoveries made by telescope.”

“The things you can prove may be just tiny islands, exceptions, compared to the vast sea of results that cannot be proved by human thought alone.”

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In conclusion ...

To commemorate the proof, the University of Illinois adopted this postage meter stamp:

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Thank you