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Today:
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Reading:
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Questions (e.g., on HW #4)?
Problem 4.1.9
Information about Exam #2
Planar Graphs
[CH] 5.1-5.2
[HR] 8.1
Exercises:
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[CH] p. 160: 5.1.1, 5.1.5;
[CH] p. 167: 5.2.1, 5.2.4, 5.2.5
[HR] p. 154: 8.1.1, 8.1.2
Thursday, 10/31/13, Slide #1
The Icosian Game

From http://puzzlemuseum.com/month/picm02/200207icosian.htm:

An original copy of Sir William Rowan Hamilton's famous "Icosian Game". There are
only three other known examples of this puzzle. Sir William Rowan Hamilton, who
was Astronomer Royal of Ireland, invented the puzzle in 1857. He sold the
rights to Jaques for £25. Hamilton gave his name to the mathematical field of
"Hamiltonian Circuits". The pieces are conical bone or ivory plugs. The puzzle museum
only has a photocopy copy of the original rules, which give 15 example puzzles. Hamilton
intended that one person should pose the puzzle and a second person solve it.
Note: Graph is a
Dodecahedron
(not Icosahedron)
Thursday, 10/31/13, Slide #2
Exam 1: Next Thursday, 11/7/13
(17.5% of course grade)

Topics: Everything since Exam #1, up to and
including today’s class:
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Types of Questions:
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State definitions, theorems carefully
Give examples and counter-examples
Solve problems similar to homework and examples from class and
text
Reference Sheet:
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All topics covered in class and related readings, all exercises, all
hand-in homework
You may bring one reference sheet, 8.5” x 11”
OK to write anything you want, on both sides, in any size font.
Bring questions for exam to class on Tuesday!
Thursday, 10/31/13, Slide #3
Material for Exam #2
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[CH] 2.5-2.6
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Breadth-First Search
Depth-First Search
Dijkstra’s Algorithm
Cut vertices and bridges
Separating sets
-connectivity and
Whitney’s Theorem
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Euler Circuits and Trails
Characterizations of Eulerian graphs:
connected & all even degrees;
connected & union of edge-disjoint
cycles
Fleury’s Algorithm
Chinese Postman Problem
Hamiltonian cycles and paths
Necessary conditions
Dirac’s and Ore’s Theorems
Closure of a graph
Traveling Salesperson Problem (TSP)
Lower bounds for TSP
Nearest Neighbor and Cheapest Link
Algorithms
Matching problems
Saturated and unsaturated vertices
Maximum and perfect matchings
Alternating and Augmenting Paths
Berge’s Theorem
Maximum Matching Algorithm
Hall’s Marriage Theorem
Perfect matchings in -regular bipartite
graphs
Vertex Covers
Konig’s Theorem
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[CH] 3.1-3.4
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[CH] 4.1-4.3, 4.5
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[CH] 5.1-5.2 (up to what is covered in
class today)
Plane graphs and planar graphs
Jordan Curve Theorem
Non-planarity of
and
,
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Faces of plane graphs
Euler’s Formula
Edge bounds for planar graphs
Low-degree vertices in planar graphs
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Thursday, 10/31/13, Slide #4
Planar Graphs
A plane drawing of a graph is a drawing of the graph
in the plane with no edge crossings
 A planar graph is a graph that can be drawn in the
plane, i.e., has a planar drawing
 A plane graph is a particular plane drawing of a
planar graph.
To prove a graph is planar: one way is to draw it!
 Examples: Prove that each of the following graphs is
planar:
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The complete graphs
, , ,
The complete bipartite graphs
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Any tree or cycle
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,
and
,
for any
1
Thursday, 10/31/13, Slide #5
Proving a graph is non-planar using the
Jordan Curve Theorem

Jordan Curve Theorem: Any simple closed
curve C in the plane divides the plane into
two disjoint regions: the inside and the
outside.
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Any curve joining an inside point to an outside
point must intersect the curve C.
Using JCT to prove nonplanarity of
contradiction:


by
Assume has a plane drawing. Then any drawing
of G contains a drawing of any cycle of G as a
simple closed curve in the plane.
Add vertices and edges of G to drawing, until some
edge is forced to cross the boundary of a cycle.
Thursday, 10/31/13, Slide #6
Nonplanarity of K5 and K3,3

Theorem 5.1. The complete graph
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Proof:
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is nonplanar.
Start by drawing cycle 1-2-3-1.
Next add vertex 4 and its edges
Then try to add vertex 5 and its edges.
Theorem 5.2. The complete bipartite graph
nonplanar.
 Proof: Similar to Theorem 5.1
,
is
Thursday, 10/31/13, Slide #7
Some basic results about planarity
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Theorem. A graph can be embedded (drawn
without crossings) in the plane if and only if it can be
embedded on the sphere.
Theorem. If is a planar graph, then any subgraph
of is planar.
Corollary. If has a non-planar subgraph, then is
non-planar.
Corollary. If has either
or
as a subgraph,
,
then is non-planar.
 If , the complete graph
is non-planar.
 If and , the complete bipartite graph
, is non-planar.
Thursday, 10/31/13, Slide #8
Faces of plane graphs
([CH] 5.2, [HR] 8.1)
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A face of a plane graph G is a set of points each pair
of which can be connected by a curve that crosses no
edge of G
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The outer face is called
the exterior face
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and
are in the same face, but not
On the sphere, there is no
exterior face!
The boundary of a face
is the closed walk around
its border
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Center face has boundary
6-7-8-5-11-5-6
Thursday, 10/31/13, Slide #9
Euler’s Formula
One of the three most important results on planar
graphs: An algebraic relationship between the
numbers of vertices, edges, and faces of a plane
graph:

Euler’s Formula. If is a connected,
plane graph and , ,and are the numbers
of vertices, edges, and faces of , then
–  Example. Check this for the graph on
previous page;
; a tree with vertices
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Thursday, 10/31/13, Slide #10
Corollaries

Corollary 1. Any two plane drawings of a
planar graph have the same number of faces.
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Corollary 2. If is a simple plane graph with at
least 3 edges, then
– .
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Important, but not obvious! We’ll prove this.
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Example: Use Cor. 2 to prove
about
?
,
is non-planar. What
Corollary 3. If is a simple planar graph, then
has a vertex with degree at most 5.
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Follows from Cor. 2 and Degree-Sum Theorem. We’ll
prove this too!
Thursday, 10/31/13, Slide #11
Proof of Euler’s Formula
Proof by induction on the number of
faces:
 Base case: f = 1. What is G?
 Inductive case: Suppose formula is true
for all graphs with at most k faces, and
suppose G has k + 1 faces
 Then G has a cycle – why?
 Look at what happens if we remove a
cycle edge from G
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Thursday, 10/31/13, Slide #12