Drawing of G

Drawing of G

Planar Embedding of G

Chord

• A chord of a cycle C is an edge not in C whose endpoints lie in C

Proposition 6.1.2

Proof. 1. Consider a drawing of K5 or K3,3 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.

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 ConflictTwo Chords Conflict

Red Line : chord

Proposition 6.1.2

5. K3,3 has three pairwise conflict chords.

We can put at most inside and one outside, so it is not possible to complete the embedding.

Proposition 6.1.2

5. In K5, 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

Faces

Definition 6.1.11

1

2

3

4

5

6

7

Example 6.1.12

L(F2)=6L(F0)=7 L(F1)=3 L(F2)=9L(F0)=4 L(F1)=3

Cut edge

F0

F1 F2 F0F1

F2

Proposition 6.1.13

Euler’s Formula

Euler’s Formula

Theorem 6.1.23

Nonplanarity of K5 and K3,3

K5 (e = 10, n = 5) K3,3 (e = 9, n = 6)