A Simple Proof of the Four-Color Theorem
  A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete graph. Then it is shown that for the theorem to be false there must exist a complete planar graph of $h$ edges such that $h>4$. Finally the theorem is proved to be true by showing that there does not exist a complete planar graph with $h>4$. 
Fayez A. Alhargan