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With the help of Neutrosophy and Quad-stage Method, the proof for negation of "the four color theorem" is given. In which the key issue is to consider the color of the boundary, thus "the two color theorem" and "the five color theorem" are derived to replace "the four color theorem".doi:10.5281/zenodo.22571 fatcat:ly7akjrje5afliaou7mqzf3e54