Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Four Color Theorem Explained
1. The Four Color
Theorem
For any subdivision of the plane into nonoverlapping regions, it is always possible
to mark each of the regions with four
different colors in such a way that no two
adjacent regions receive the same color .
2. What is the Four Color Theorem?
Given any separation of a plane
into adjacent regions , the regions
can be colored using at most four
colors so that no two adjoining
regions have the same colour
A region is adjacent to the other
region if they share a boundary
and not just a point.
3. HISTORY
The four color theorem was proven in 1976 by Kenneth Appel and
Wolfgang Haken. It was the first major theorem to be proved using a
computer. Appel and Haken's approach started by showing that there is a
particular set of 1,936 maps, each of which cannot be part of a smallestsized counterexample to the four color theorem. Appel and Haken used a
special-purpose computer program to confirm that each of these maps
had this property. Additionally, any map (regardless of whether it is a
counterexample or not) must have a portion that looks like one of these
1,936 maps. Showing this required hundreds of pages of hand analysis.
Appel and Haken concluded that no smallest counterexamples existed
because any must contain, yet not contain, one of these 1,936 maps. This
contradiction means there are no counterexamples at all and that the
theorem is therefore true. Initially, their proof was not accepted by all
mathematicians because the computer-assisted proof was infeasible for a
human to check by hand (Swart 1980). Since then the proof has gained
wider acceptance, although doubts remain (Wilson 2002, 216–222).
To dispel remaining doubt about the Appel–Haken proof, a simpler proof
using the same ideas and still relying on computers was published in 1997
by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the
theorem was proven by Georges Gonthier with general purpose theorem
proving software.
4. Why it took so long to prove the
Theorem?
The Four Color Problem
It was easy to prove that 6 colors would
work and reduce that numbers to 5 but
proving that 4 colors would work took
longer and many counterexamples
exist.
Most counter examples try to have one
region that is adjacent to all other
regions.
No way to check with actually coloring
the maps.
5. Proving the Four Color
Theorem
The proof relies on being able to reduce a set of configurations on a graph.
This small configuration of the original can be easily colored with four colors.
Many times the figures can be colored by focusing on one region and trying
to color the figures around it with the other colors.
To prove the (network version of the) Four Color Theorem, you start out by
assuming that there is a network that cannot be colored with four colors, and
work to deduce a contradiction. If there is such a network, there will be (at
least) one that has the fewest number of nodes. That's the one to look at.
The idea then is to show that you can find a particular node that can be
removed without altering the number of colors needed to color the network.
Since that new network has one fewer nodes than the one you started with,
and that initial network was chosen to be the smallest that could not be
colored with four colors, the new network can be colored with four colors.
But then, because of the way you chose the node to remove, that means the
original map can be colored with four colors. And there's the contradiction.
