Spiral Chains: A New Proof of the Four Color Theorem

Abstract

Acceptable but due to extensive usage of a computer rather unpleasant proof of the famous four color map problem of Francis Guthrie were settled eventually by W. Appel and K. Haken in 1976. Using the same method but shortening the proof twenty years later by another team, namely N. Robertson, D.P. Sanders, P.D. Seymour and R. Thomas would not improve considerably the readability of the proof either. Thus it has been widely accepted the need of more elegant and readable proof. There are considerable number of equivalent formulations of the problem but none of them promising for a possible non-computer proofs. On the other hand known proofs are used the concept of Kempe chain and reducibility of the configurations which were a century old ideas. With these in mind we have introduced a new concept which we call "spiral chains" in the maximal planar graphs. We have shown that for any maximal graph as long as spiral chains are being used we do not need the fifth color. Henceforth this paper offers another proof to the four color theorem which is not based on deep and abstract theories from the other branches of mathematics or using computing power of computers, but rather completely on a new idea in graph theory.