FOUR COLOR THEOREM
The four-color theorem states that any map in
a plane can be colored using four-colors in
such a way that regions sharing a common
boundary (other than a single point) do not
share the same color.
A BRIEF HISTORY
• Francis Guthrie created the four color problem while
coloring a map of England while in law school
• Francis told his brother the problem and his brother
gave the problem to his professor, De Morgan.
• De Morgan spread the problem across the
mathematical world and eventually taught the
problem to Cayley
• Cayley taught the problem to Kempe, who used
Kempe chains to give the first proof of the problem
A BRIEF HISTORY
• Kempe’s proof was found to have an error in it. Later
Appel and Haken finnally came up with the final proof
with the help of computers.
• The Four Color Theorem is the 1st theorem proven by a
computer.
KEMPE’S CHAINS
Let G be a graph with a coloring using at least two
different colors represented by i and j. Let H(i, j) denote
the subgraph of G induced by all the vertices of G
colored either i or j and let K be a connected component
of the subgraph H(i, j). If we interchange the colors i and j
on the vertices of K and keep the colors of all other
vertices of G unchanged, then we get a new coloring of
G, which uses the same colors with which we started.
PROOF
• Step 1: Let G be a non 4-colorable planar graph
•
Step 2: prove that G has at least one of 1476 configurations that are
unavoidable.
•
Step 3: Prove that those 1476 configurations are reducible
•
Step 4: There is no such thing as a minimal non 4-colorable planar
graph. Therefore, there are no non-4-colorable planar graphs.
2 KEY PIECES
The discharging method: used to prove
that every graph in a certain class
contains some subgraph from a
specified list.
Reducibility: finding unavoidable sets