1. Drawing of G

3. Planar Embedding of G

4. Chord A chord of a cycle C is an edge not in C whose endpoints lie in C

5. Proposition 6.1.2 Proof. 1. Consider a drawing of K 5 or K 3,3 in the plane. Let C be a spanning cycle. 2. If the drawing does not have crossing edges, then C is drawn as a closed curve.

6. Proposition 6.1.2 3. Two chords conflict if their endpoints on C occur in alternating order. 4. When two chords conflict, we can draw only one inside C and one outside C. Two Chords do not Conflict Two Chords Conflict Red Line : chord 

7. Proposition 6.1.2 5. K 3,3 has three pairwise conflict chords. We can put at most inside and one outside, so it is not possible to complete the embedding.

8. Proposition 6.1.2 5. In K 5, at most two chords can go outside or inside. Since there are five chords, it is not possible to complete the embedding. Red Line : chord

9. Faces

10. Definition 6.1.11 1 2 3 4 5 6 7

11. Example 6.1.12 L(F 2 )=6L(F 0 )=7L(F 1 )=3 L(F 2 )=9L(F 0 )=4L(F 1 )=3 Cut edge F0F0 F1F1 F2F2 F0F0 F1F1 F2F2

12. Proposition 6.1.13

13. Euler’s Formula

15. Theorem 6.1.23

16. Nonplanarity of K 5 and K 3,3 K 5 (e = 10, n = 5)K 3,3 (e = 9, n = 6)