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

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

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

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

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

8. Sec. 5.1: Planarity & Coloring
Planar Graph Example: E A B C D
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. Sec. 5.1: Planarity & Coloring
Planar Graph Example:
Now do problems 1-4 on pp

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

12. Sec. 5.1: Planarity & Coloring
Bipartite Graph Example: A
The vertices of this graph can be divided into two distinct sets: (chromatic number = 2) B C
{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
Subgraph Example:
Now do problems 5-6 on p. 218.

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

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