Abstract

By using the color exchange method and convertible property of the topological graph for inside cycle and outside, the colors
of the vertexes on the two cycles in the same position can be corresponded with each other. Thus prove the reducibility
of the 3-cycle and 4-cycle with both vertexes inside and outside. Meanwhile, point out the invalid counterexample of the
“Heawood graph” which was commonly accepted in many textbooks.

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