Sec. 5.1: Planarity & Coloring

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Presentation transcript:

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

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

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

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:

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

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

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.

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.

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

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

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

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

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

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

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

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.

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

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