MATH 310, FALL 2003
(Combinatorial Problem
Solving)
Lecture 4, Monday, September 8
1.4. Planar Graphs

Homework (MATH 310#2M):
• Read 2.1. Read Appendix A.2. Write down a list of all
newly introduced terms (printed in boldface or italic)
• Do Exercises1.4: 3,6,14,18,20,24,27
• Volunteers:
• ____________
• ____________
• Problem: 18.

News: There is a “Download Directory”
on the class webpage.
Planar and Plane Graphs


(a)
(b)

(c)
(d)
A graph is planar if it
can be drawn without
edges crossing.
The term plane graph
refers to a planar
depiction of a planar
graph.
(a) is planar,(b),(c)
are not. (d) is plane.
Dual Graph



Normally a vetrex
is also included for
the unbounded
region.
Warning: There are
maps with nonsimple duals!
Instead of coloring
regions of the
plane graph we
may color vertices
of its dual.
Circle-Chord Method
1.
2.
3.
Find a circuit that contains all vertices
and draw it as a large circle. [Give up, if
there is no such circuit].
Draw the remaining edges either in the
circle or outside the circle.
We either finish by drawing the graph
successfuly or we get stuck and the
graph is non-planar.
Complete Bipartite Graph
Km,n.


Km,n is a complete
bipartite graph
consisting of a set
with m vertices and a
set with n vertices
with each vertex in
one set adjacent to all
vertices in the other
set.
The graph on the left
is K3,3. Show by
circle-chord method
that it is non-planar.
K5 is non-planar.

By circle-chord
method me may
prove that K5 is
non-planar.
G-Configuration



G-configuration is any
graph that is obtained
from G by adding
some vertices in the
middle of some
edges.
We are mainly
interested in K3,3 and
K5 configurations.
On the left we see a
K3,3-configuration.
Theorem 1 (Kuratowski,
1930)

A graph is planar if and only if it
does not contain a subgraph that is
a K5 or K3,3 configuration.
Notation

For plane graphs we
use the following
notation:




v = # vertices
e = # edges
r = # regions
(including the
unbounded region)
The graph on the left
has v = 8, e = 12, r =
6.
Theorem 2 (Euler, 1752)

If G is a connected planar graph,
then any plane graph depiction of G
has
• r = e - v +2.

Proof: By mathematical induction.
[also called induction method,
principle of induction, ...]
Combinatorial “principles”.

So far we have encountered two
methods that we call principles:
• The bookkeeper’s principle
• The induction principle

We will learn several other useful
principles that help proving
combinatorial results and solving
combinatorial problems.
Example 5: using Euler’s
Formula
How many regions would be in a
plane graph with 10 vertices each of
degree 3?
 Answer: 7.

Corollary

If G is a connected
planar graph with
e > 1, then

• e · 3v – 6.
Warning: There
are non-planr
graphs that satisfy
the condition
• e · 3v – 6.

For example, take
K3,3.
Example 6: K5.

The graph K5 is
non-planar by
Euler’s Formula.