1. MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 4, Monday, September 8

2. 1.4. Planar Graphs Homework (MATH 310#2M): Read 2.1. Read Appendix A.2. Write down a list of all newly introduced terms (printed in boldface or italic) Do Exercises1.4: 3,6,14,18,20,24,27 Volunteers: ____________ Problem: 18. News: There is a “Download Directory” on the class webpage. News: There is a “Download Directory” on the class webpage.

3. Planar and Plane Graphs A graph is planar if it can be drawn without edges crossing. The term plane graph refers to a planar depiction of a planar graph. (a) is planar,(b),(c) are not. (d) is plane. (a) (b) (c)(d)

4. Dual Graph Normally a vetrex is also included for the unbounded region. Warning: There are maps with non- simple duals! Instead of coloring regions of the plane graph we may color vertices of its dual.

5. Circle-Chord Method 1. Find a circuit that contains all vertices and draw it as a large circle. [Give up, if there is no such circuit]. 2. Draw the remaining edges either in the circle or outside the circle. 3. We either finish by drawing the graph successfuly or we get stuck and the graph is non-planar.

6. Complete Bipartite Graph K m,n. K m,n is a complete bipartite graph consisting of a set with m vertices and a set with n vertices with each vertex in one set adjacent to all vertices in the other set. The graph on the left is K 3,3. Show by circle-chord method that it is non-planar.

7. K 5 is non-planar. By circle-chord method me may prove that K 5 is non-planar.

8. G-Configuration G-configuration is any graph that is obtained from G by adding some vertices in the middle of some edges. We are mainly interested in K 3,3 and K 5 configurations. On the left we see a K 3,3 -configuration.

9. Theorem 1 (Kuratowski, 1930) A graph is planar if and only if it does not contain a subgraph that is a K 5 or K 3,3 configuration.

10. Notation For plane graphs we use the following notation: v = # vertices e = # edges r = # regions (including the unbounded region) The graph on the left has v = 8, e = 12, r = 6.

11. Theorem 2 (Euler, 1752) If G is a connected planar graph, then any plane graph depiction of G has r = e - v +2. Proof: By mathematical induction. [also called induction method, principle of induction,...]

12. Combinatorial “principles”. So far we have encountered two methods that we call principles: The bookkeeper’s principle The induction principle We will learn several other useful principles that help proving combinatorial results and solving combinatorial problems.

13. Example 5: using Euler’s Formula How many regions would be in a plane graph with 10 vertices each of degree 3? Answer: 7.

14. Corollary If G is a connected planar graph with e > 1, then e · 3v – 6. Warning: There are non-planr graphs that satisfy the condition e · 3v – 6. For example, take K 3,3.

15. Example 6: K 5. The graph K 5 is non-planar by Euler’s Formula.