The Mathematics of Map
Coloring
–
The Four-Color Theorem
Roger House
Scientific Buzz Café
French Garden
Sebastopol, CA
2013 Jan 17
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.

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.
We will sketch an informal proof of this.
For a map drawn on a doughnut, things are different:
The Euler Characteristic on a torus is zero.
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F + V – E = 2
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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
Each face has at least 6 edges, so 6F ≤ E.
But each edge is counted twice, because there is a face on each side of the edge.
So 3F ≤ E.
Each vertex has at least 3 edges incident to it, so 3V ≤ E.
But each edge is counted twice, once for each end of the edge.
So 3V/2 ≤ E, or 3V ≤ 2E.
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5 or fewer neighbors
3F ≤ E
3V ≤ 2E
3F + 3V ≤ E + 2E = 3E
F + V ≤ E
F + V – E ≤ E – E = 0
F + V – E ≤ 0
What's wrong with this?
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5 or fewer neighbors
We just proved that
F + V – E = 2 so it cannot be that
F + V – E ≤ 0
Thus the assumption 'every face has at least
6 neighbors' must be false.
So, at least one face has 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.
From the set of such maps pick one with the fewest countries.
By the theorem we just proved, one country
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Six Colors Suffice
Here's such a country (whose neighbors may be more complicated than shown).
We remove one edge:
How many colors are needed?
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Six Colors Suffice
The new map can be colored with 6 colors because it has fewer countries than a minimal map requiring more than 6 colors.
This part of the map requires at most 5 colors:
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Six Colors Suffice
Add back the edge we removed.
Use the sixth color.
We have colored the map with 6 colors.
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Six Colors Suffice
Wait a minute!
We started with a map which required more than 6 colors.
But we have just shown how to color it with
6 colors.
This is a contradiction.
Which proves that all maps require no more than 6 colors. █
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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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Break time
While on break, you might try coloring the following map to see if it requires 4, 5, or 6 colors:
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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?
We will give a rough idea of Kempe's approach.
We won't examine a lot of picky cases which the actual proof dealt with.
We'll focus on example maps, not the general case.
But, you'll get the idea ...
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Is this approach familiar?
Assume that some maps exist which require more than four colors.
From these maps, pick one that has the fewest number of countries.
There may be more than one such map.
Never mind – pick any map from the set of those with the fewest countries.
Say this map:
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Minimum map for which 4 colors are not enough
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Get rid of one country
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Now we can 4-color the map
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Restore the removed country
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Change gray to what?
We want to color the gray country red, green, blue, or yellow.
In order to do this, we must change the colors of some of the countries bordering the gray country.
Consider a red-yellow chain:
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Red-yellow chain
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Swap B-G inside R-Y chain
`
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Red-green chain
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Swap B-Y inside R-G chain
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Color the gray country blue
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Here's a 4-coloring of the map
What's wrong with this picture?
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Four Colors Suffice!
The map requires more than 4 colors.
But we have just shown that it can be colored with 4 colors .
Contradiction!
So the theorem is proved: There are no maps which need more than 4 colors. █
Ah (a sigh of relief), all is right with the world.
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Hold it a second
We haven't actually proved the theorem.
We've looked at one example map and showed how Kempe argued that it could be 4-colored, using Kempe chains.
His actual proof dealt with a number of cases which we've ignored.
It dealt with general cases, not examples.
But we've seen the main idea of his method.
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Trouble in paradise
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 Mapcolour theorem .
The paper contained this map:
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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 an example?
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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