  The famous four color theorem states that for all planar graphs, every vertex
can be assigned one of 4 colors such that no two adjacent vertices receive the
same color. Since Francis Guthrie first conjectured it in 1852, it is until
1976 with electronic computer that Appel and Haken first gave a proof by
finding and verifying 1936 reducible unavoidable sets, and a simplified proof
of Robertson, Sanders, Seymour and Thomas in 1997 only involved 633 reducible
unavoidable sets, both proofs could not be realized effectively by hand. Until
now, finding the reducible unavoidable sets remains the only successful method
to use, which came from Kempe's first "proof" of the four color problem in
1879. An alternative method only involving 4 reducible unavoidable sets for
proving the four color theorem is used in this paper, which takes form of
mathematical proof rather than a computer-assisted proof and proves both the
four color conjecture and the uniquely 4-colorable planar graph conjecture by
mathematical method.
