Today: Reading: Exercisesadean/MC3021309/ScreenSlides/MC302_131031_S.pdfThursday, 10/31/13, Slide #2 The Icosian Game ... Chinese Postman Problem

Thursday, 10/31/13, Slide #1

Today: Questions (e.g., on HW #4)? Problem 4.1.9 Information about Exam #2 Planar Graphs

Reading: [CH] 5.1-5.2 [HR] 8.1

Exercises: [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


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)

http://puzzlemuseum.com/month/picm02/200207icosian.htm


Exam 1: Next Thursday, 11/7/13 (17.5% of course grade) Topics: Everything since Exam #1, up to and

including today’s class: All topics covered in class and related readings, all exercises, all

hand-in homework Types of Questions:

State definitions, theorems carefully Give examples and counter-examples Solve problems similar to homework and examples from class and

text Reference Sheet:

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!


Material for Exam #2 [CH] 2.5-2.6

Breadth-First Search Depth-First Search Dijkstra’s Algorithm Cut vertices and bridges Separating sets 𝒌-connectivity and 𝜿(𝑮) Whitney’s Theorem

[CH] 3.1-3.4 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

[CH] 4.1-4.3, 4.5 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 𝒌-regular bipartite

graphs Vertex Covers Konig’s Theorem

[CH] 5.1-5.2 (up to what is covered in class today) Plane graphs and planar graphs Jordan Curve Theorem Non-planarity of 𝑲𝟓 and 𝑲𝟑,𝟑 Faces of plane graphs Euler’s Formula Edge bounds for planar graphs Low-degree vertices in planar graphs


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: The complete graphs 𝑲𝟏,𝑲𝟐,𝑲𝟑,𝑲𝟒 The complete bipartite graphs 𝑲𝟏,𝒎 and 𝑲𝟐,𝒎 for any 𝑚 ≥ 1 Any tree or cycle


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. Any curve joining an inside point to an outside

point must intersect the curve C. Using JCT to prove nonplanarity of 𝐺 by

contradiction: Assume 𝐺 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.


Nonplanarity of K5 and K3,3 Theorem 5.1. The complete graph 𝑲𝟓 is nonplanar. Proof:

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 𝑲𝟑, 𝟑 is

nonplanar. Proof: Similar to Theorem 5.1


Some basic results about planarity Theorem. A graph 𝑮 can be embedded (drawn

without crossings) in the plane if and only if it can be embedded on the sphere.

Theorem. If 𝑮 is a planar graph, then any subgraph of 𝑮 is planar.

Corollary. If 𝑮 has a non-planar subgraph, then 𝑮 is non-planar.

Corollary. If 𝑮 has either 𝑲𝟓 or 𝑲𝟑, 𝟑 as a subgraph,

then 𝑮 is non-planar. If 𝒏 ≥ 𝟓, the complete graph 𝑲𝒏 is non-planar. If 𝒑 ≥ 𝟑 and 𝒒 ≥ 𝟑, the complete bipartite graph 𝑲𝒑,𝒒 is non-planar.


Faces of plane graphs ([CH] 5.2, [HR] 8.1) 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 𝒙 and 𝒚 are in the same face, but not 𝒛

The outer face is called the exterior face On the sphere, there is no

exterior face! The boundary of a face

is the closed walk around its border Center face has boundary

6-7-8-5-11-5-6


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 𝑮 is a connected, plane graph and 𝒏, 𝒆, and 𝒇 are the numbers of vertices, edges, and faces of 𝑮, then 𝒏 – 𝒆 + 𝒇 = 𝟐

Example. Check this for the graph on previous page; 𝑲𝟒; a tree with 𝒏 vertices


Corollaries Corollary 1. Any two plane drawings of a

planar graph have the same number of faces. Corollary 2. If 𝑮 is a simple plane graph with at

least 3 edges, then 𝒆 ≤ 𝟑 𝒏 – 𝟔. Important, but not obvious! We’ll prove this. Example: Use Cor. 2 to prove 𝑲𝟓 is non-planar. What

about 𝑲𝟑, 𝟑?

Corollary 3. If 𝑮 is a simple planar graph, then 𝑮 has a vertex with degree at most 5. Follows from Cor. 2 and Degree-Sum Theorem. We’ll

prove this too!


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

Today: Reading: Exercisesadean/MC3021309/ScreenSlides/MC302_131031_S.pdfThursday, 10/31/13, Slide #2 The Icosian Game ... Chinese Postman Problem

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