Graph Planarity and Colorability

1
Graph Planarity and Colorability
Homework 9.7 6, 8, 12, 20, 22, 24 9.8 2, 4,
6, 8, 20
2
Three house/ three utility problem

• Is it possible to join three houses to three
  separate utilities so that none of the
  connections cross?

3
Planar graphs

• A graph is planar if it can be drawn in the
  plane without any edges crossing

4
Eulers formula for planar graphs

• Theorem Let G be a connected planar graph with
  v vertices, e edges and r regions formed drawing
  G in the plane with no edge crossing. Then
• v e r 2

5
K3,3 is not planar

• Consider region R formed by vertices 1, 2, 4, 5.
• Case 1 vertex 3 inside R
• Case 2 vertex 3 outside R

Where can vertex 6 go?
2
1
3
6
4
5
6
Kuratowskis Theorem

• Are there nonplanar graphs other than K3,3 and
  K5?
• If a graph has K3,3 or K5 as a subgraph
• If a graph has a subgraph that can be obtained
  by adding intermediate vertices to edges of
  either K3,3 or K5 (homeomorphic)
• any others?
• Kuratowskis Theorem
• A graph is nonplanar if and only if it has a
  subgraph homeomorphic to either K3,3 or K5.

7
Graph Coloring

• What is the least number of colors needed to
  color a map?

8
Dual Map
Region ? vertex Common border ? edge
G
Chromatic number ?(G)
What is the least number of colors needed to
color the vertices of a graph so that no two
adjacent vertices are assigned the same color?
9
Four color theorem
Every planar graph is 4-colorable
10
What about non-planar graphs?
K5
K3,3
11
Chromatic Numbers of Some Graphs

• ?(G) 1 iff ...
• For Kn, the complete graph with n vertices, ?(Kn)
  Corollary If a graph has Kn as its subgraph,
  then ?(Kn)
• For Cn, the cycle with n vertices, ?(Cn)
• For any bipartite graph G ?(G)
• For any planar graph G, ?(G) 4 (Four Color
  Theorem)

12
Chromatic Numbers (cont.)

• There is no efficient algorithm for finding ?(G)
  for an arbitrary graph. Most computer scientists
  believe that such an algorithm doesnt exist.
• There is an efficient algorithm to determine
  whether a given graph is bipartite, i.e.,
  2-colorable.
• Applications
• map coloring
• scheduling