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

This lecture introduces planar graphs and explores their properties. Topics include dual graphs, circle-chord method, complete bipartite graphs, G-configurations, Kuratowski's theorem, Euler's formula, and combinatorial principles.

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

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  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.

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

  6. Complete Bipartite Graph Km,n. • Km,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 K3,3. Show by circle-chord method that it is non-planar.

  7. K5 is non-planar. • By circle-chord method me may prove that K5 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 K3,3 and K5 configurations. • On the left we see a K3,3-configuration.

  9. Theorem 1 (Kuratowski, 1930) • A graph is planar if and only if it does not contain a subgraph that is a K5 or K3,3 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. 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 K3,3. Corollary

  15. The graph K5 is non-planar by Euler’s Formula. Example 6: K5.

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