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
D
C
AC intersects BD.
Sec. 5.1: Planarity & Coloring
Planar Graph Example:
A
B
D
C
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
D
C
A
B
D
C
Sec. 5.1: Planarity & Coloring
Planar Graph Example:
E
A
B
D
C
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
D
C
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
B
C
D
E
The vertices of this graph
can be divided into two
distinct sets: (chromatic
number = 2)
Sec. 5.1: Planarity & Coloring
Bipartite Graph Example:
A
B
C
D
E
The vertices of this graph
can be divided into two
distinct sets: (chromatic
number = 2)
{A, C, E}
{B, D}
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:
A
B
A
B
C
D
C
D
Sec. 5.1: Planarity & Coloring
Graph Complement Example:
Now do problems 7, 9-12
on pp. 218-219.