FOUR COLOR THEOREM BY: JUSTIN GILMORE

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 1 st 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