1. 5-color theorem 林典濤 : PPT, Lecture note 陳咨翰 : Presentation

2. 4-color theorem Statement Proved by who? History

3. Why it was wrong? P.J Heawood Counterexample

4. 5-color theorem Proved by Heawood Statement: The chromatic number of a planar graph is, at most, 5.

5. Main Idea (I) Theorem 13.2.2 Let G be a connected planar graph. Then δ (G) ≦ 5.

6. Main Idea (II) Theorem 13.3.1 (page.537) Graph H HHHHHH

7. Proof of 5-coloring Theorem Theorem 13.3.2 (page.537) Let G be a planar graph of order n. We assume that G is drawn in the plane as plane-graph. (G is embedded on the plane.) If n ≦ 5, then X(G) ≦ 5. (Trival) Let n>5 and use induction By Thm(13.2.2), there is a vertex y whose degree at most 5 Let H = G – y, and order of H is n-1, by the induction hypothesis, H has a 5-coloring. If the degree of y is 4 or less, then we assign one of 5 colors to y and obtain a 5-coloring of G. Suppose the degree of y is 5. There are 5 vertices adjacent to y.

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