The Mathematics of Map Coloring – The Four-Color Theorem

The Problem

Counties of England

The Counties, Colored

Why Counties of England?

1852 October 23

How to begin?

1 country – how many colors?

1 country – 1 color

2 countries – how many colors?

2 countries – 2 colors

3 countries – how many colors?

3 countries – 3 colors

3 countries – how many colors?

3 countries – 2 colors

4 countries – how many colors?

4 countries – 2 colors

4 countries – how many colors?

4 countries – 3 colors

4 countries – how many colors?

4 countries – 4 colors

Important fact

Do the math

Leonhard Euler (1707 -83)

F+V–E=?

Euler Characteristic

F+V–E=2

5 or fewer neighbors

5 or fewer neighbors

5 or fewer neighbors

5 or fewer neighbors

Six Colors Suffice

Six Colors Suffice

Six Colors Suffice

Six Colors Suffice

Six Colors Suffice

Pick a number: 4, 5, or 6?

Break time

No activity for several decades

Solved, at last

How did he do it?

Is this approach familiar?

Minimum map for which 4 colors are not enough

Get rid of one country

Now we can 4 -color the map

Restore the removed country

Change gray to what?

Red-yellow chain

Swap B-G inside R-Y chain

Red-green chain

Swap B-Y inside R-G chain

Color the gray country blue

Here's a 4 -coloring of the map

Four Colors Suffice!

Hold it a second

Trouble in paradise

Ouch!

The Five Color Theorem

Maybe 4 colors aren't enough?

Gardner: 5 colors are needed

80+ years had gone by. . .

Garner's map: 4 colors work

At last!

Some reactions

What's the problem?

Eventually: Acceptance

In conclusion. . .