jordan curve theorem a simple closed curve cuts its interior from its exterior

Jordan Curve Theorem

A simple closed curve cuts its interior from its exterior.

Theorem 6.3.1

Theorem 6.3.1

Theorem 6.3.1

Dual GraphThe dual graph G* if a plane graph is a plane

graph whose vertices corresponding to the faces of G. The edges of G* corresponds to the edges of G as follows: if e is an edge of G with face X on one side and face Y on the other side, then the endpoints of the dual edge e* in E(G*) are the vertices x and y of G* that represents the faces X and Y of G.

K4

Proper Face-Coloring

Proper 3-edge-Coloring

Theorem 7.3.2

Theorem 7.3.2

Theorem 7.3.2

Theorem 7.3.2

Theorem 7.3.2

Theorem 7.3.4

Tait Coloring

Tait’s Conjecture

Grinberg’s Sufficient Condition

Grinberg’s Condition

Grinberg’s Condition

Example 7.3.6

Example 7.3.6

Example 7.3.6

Are Planar Graph 4-Colorable?

• The four color theorem was proven using a computer, and the proof is not accepted by all mathematicians because it would be unfeasible for a human to verify by hand. Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof. (see http://en.wikipedia.org/wiki/Four_color_theorem)

• See pages 258-260 in the text book.