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The famous four color theorem states that for all planar graphs, every vertexcan be assigned one of 4 colors such that no two adjacent vertices receive thesame color. Since Francis Guthrie first conjectured it in 1852, it is until1976 with electronic computer that Appel and Haken first gave a proof byfinding and verifying 1936 reducible unavoidable sets, and a simplified proofof Robertson, Sanders, Seymour and Thomas in 1997 only involved 633 reducibleunavoidable sets, both proofs could not be realized effectively by hand. Untilnow, finding the reducible unavoidable sets remains the only successful methodto use, which came from Kempe's first "proof" of the four color problem in1879. An alternative method only involving 4 reducible unavoidable sets forproving the four color theorem is used in this paper, which takes form ofmathematical proof rather than a computer-assisted proof and proves both thefour color conjecture and the uniquely 4-colorable planar graph conjecture bymathematical method.

Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph