Sec. 5.1: Planarity & Coloring

1. Sec. 5.1: Planarity & Coloring Key Terms: Planar Graph Bipartite Graph Subgraph Complement of a Graph

2. Sec. 5.1: Planarity & Coloring Key Terms: Planar Graph—A graph is planar if it can be drawn in such a way that edges intersect only at vertices. Bipartite Graph Subgraph Complement of a Graph

3. A B D C Sec. 5.1: Planarity & Coloring Planar Graph Example: AC intersects BD.

4. A B D C Sec. 5.1: Planarity & Coloring Planar Graph Example: AC intersects BD. But we can redraw the graph so that they don’t intersect:

5. A B D C Sec. 5.1: Planarity & Coloring Planar Graph Example: A B D C

6. E A B D C Sec. 5.1: Planarity & Coloring Planar Graph Example: Can you redraw this graph with edges intersecting only at vertices?

7. Sec. 5.1: Planarity & Coloring Any planar graph has a maximum chromatic number of four. If a graph has chromatic number greater than four, it is not planar. Planar Graph Example:

8. E A B D C Sec. 5.1: Planarity & Coloring Planar Graph Example: Note that this is a K5 graph, which is not planar. This means we cannot draw a map with five countries that all border each other.

9. Now do problems 1-4 on pp. 217-218. Sec. 5.1: Planarity & Coloring Planar Graph Example:

10. Sec. 5.1: Planarity & Coloring Key Terms: Planar Graph Bipartite Graph: The vertices of a bipartite graph can be divided into two parts, or sets, such that each edge contains one vertex from each set. Subgraph Complement of a Graph

11. A Sec. 5.1: Planarity & Coloring The vertices of this graph can be divided into two distinct sets: (chromatic number = 2) B C Bipartite Graph Example: D E

12. A Sec. 5.1: Planarity & Coloring The vertices of this graph can be divided into two distinct sets: (chromatic number = 2) B C Bipartite Graph Example: {A, C, E} {B, D} D E

13. Sec. 5.1: Planarity & Coloring Key Terms: Planar Graph Bipartite Graph Subgraph: a portion of a graph—some of the vertices and edges Complement of a Graph

14. Sec. 5.1: Planarity & Coloring Subgraph Example: The maroon graph is a subgraph of the entire graph.

15. Sec. 5.1: Planarity & Coloring Now do problems 5-6 on p. 218. Subgraph Example:

16. Sec. 5.1: Planarity & Coloring Key Terms: Planar Graph Bipartite Graph Subgraph Complement of a Graph: Any vertices that are adjacent in a graph are not adjacent in its complement, and vice-versa.

17. A B C D Sec. 5.1: Planarity & Coloring A B Graph Complement Example: C D

18. Sec. 5.1: Planarity & Coloring Now do problems 7, 9-12 on pp. 218-219. Graph Complement Example: