Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Chapter 1 Urban Services

Chi-Kwong Li

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Urban Services

Management science, also called operations research, is a study ofoptimal solutions of problems.

Here we use graph theory techniques to solve problems such as

• Parking control problem - start from one point (the controlbooth), go through all sidewalks to check the meters, andreturn to the starting point without repeating too manysidewalks.

• (Chinese) postman problem - start from one point (postoffice), deliver mail to each sidewalk, and return to thestarting point without repeating too many sidewalks.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Urban Services

Management science, also called operations research, is a study ofoptimal solutions of problems.Here we use graph theory techniques to solve problems such as

• Parking control problem - start from one point (the controlbooth), go through all sidewalks to check the meters, andreturn to the starting point without repeating too manysidewalks.

• (Chinese) postman problem - start from one point (postoffice), deliver mail to each sidewalk, and return to thestarting point without repeating too many sidewalks.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Urban Services

Management science, also called operations research, is a study ofoptimal solutions of problems.Here we use graph theory techniques to solve problems such as

• Parking control problem - start from one point (the controlbooth), go through all sidewalks to check the meters, andreturn to the starting point without repeating too manysidewalks.

• (Chinese) postman problem - start from one point (postoffice), deliver mail to each sidewalk, and return to thestarting point without repeating too many sidewalks.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Urban Services

Management science, also called operations research, is a study ofoptimal solutions of problems.Here we use graph theory techniques to solve problems such as

• Parking control problem - start from one point (the controlbooth), go through all sidewalks to check the meters, andreturn to the starting point without repeating too manysidewalks.

• (Chinese) postman problem - start from one point (postoffice), deliver mail to each sidewalk, and return to thestarting point without repeating too many sidewalks.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Seven bridge problemThe city of Königsberg in Prussia (now Kaliningrad, Russia) wasset on both sides of the Pregel River, and included two largeislands which were connected to each other and the mainland byseven bridges.

The problem was to find a walk through the city that would crosseach bridge once and only once. The islands could not be reachedby any route other than the bridges, and every bridge must havebeen crossed completely every time, which can be studied bygraph theory.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Seven bridge problemThe city of Königsberg in Prussia (now Kaliningrad, Russia) wasset on both sides of the Pregel River, and included two largeislands which were connected to each other and the mainland byseven bridges.

The problem was to find a walk through the city that would crosseach bridge once and only once. The islands could not be reachedby any route other than the bridges, and every bridge must havebeen crossed completely every time, which can be studied bygraph theory.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

The solution of Euler, 1736

• What do we need in the connectedgraph with the vertices, edges, anddegrees (valences) of vertices as shown.

• We need a circuit which covers all the edges once andexactly once. Such a circuit is called an Euler Circuit.

• If a graph has an Euler circuit,

• the graph has to be connected.• every time the circuit visits a vertex, it will take two edges -

one in and one out. So in order for the circuit cover all theedges, every vertex has to have even edges incident to it. Thatis, all the vertices have even degrees.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

The solution of Euler, 1736

• What do we need in the connectedgraph with the vertices, edges, anddegrees (valences) of vertices as shown.

• We need a circuit which covers all the edges once andexactly once. Such a circuit is called an Euler Circuit.

• If a graph has an Euler circuit,

• the graph has to be connected.• every time the circuit visits a vertex, it will take two edges -

one in and one out. So in order for the circuit cover all theedges, every vertex has to have even edges incident to it. Thatis, all the vertices have even degrees.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

The solution of Euler, 1736

• What do we need in the connectedgraph with the vertices, edges, anddegrees (valences) of vertices as shown.

• We need a circuit which covers all the edges once andexactly once. Such a circuit is called an Euler Circuit.

• If a graph has an Euler circuit,

• the graph has to be connected.• every time the circuit visits a vertex, it will take two edges -

one in and one out. So in order for the circuit cover all theedges, every vertex has to have even edges incident to it. Thatis, all the vertices have even degrees.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

The solution of Euler, 1736

• What do we need in the connectedgraph with the vertices, edges, anddegrees (valences) of vertices as shown.

• We need a circuit which covers all the edges once andexactly once. Such a circuit is called an Euler Circuit.

• If a graph has an Euler circuit,

• the graph has to be connected.

• every time the circuit visits a vertex, it will take two edges -one in and one out. So in order for the circuit cover all theedges, every vertex has to have even edges incident to it. Thatis, all the vertices have even degrees.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

The solution of Euler, 1736

• What do we need in the connectedgraph with the vertices, edges, anddegrees (valences) of vertices as shown.

• We need a circuit which covers all the edges once andexactly once. Such a circuit is called an Euler Circuit.

• If a graph has an Euler circuit,

• the graph has to be connected.• every time the circuit visits a vertex, it will take two edges -

one in and one out. So in order for the circuit cover all theedges, every vertex has to have even edges incident to it. Thatis, all the vertices have even degrees.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Leonhard Euler, 1707-1783

• Leonhard Euler was a Swiss mathematicianwho made enormous contributions to a widerange of mathematics and physics includinganalytic geometry, trigonometry, geometry,calculus, number theory and graph theory.He published over 500 works.

• A statement attributed to Pierre-Simon Laplace expressesEuler’s influence on mathematics: “Read Euler, read Euler,he is the master of us all.”

• Euler was featured on the sixth series of the Swiss 10-francbanknote and on numerous Swiss, German, and Russianpostage stamps. The asteroid 2002 Euler was named in hishonor.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Leonhard Euler, 1707-1783

• Leonhard Euler was a Swiss mathematicianwho made enormous contributions to a widerange of mathematics and physics includinganalytic geometry, trigonometry, geometry,calculus, number theory and graph theory.He published over 500 works.

• A statement attributed to Pierre-Simon Laplace expressesEuler’s influence on mathematics: “Read Euler, read Euler,he is the master of us all.”

• Euler was featured on the sixth series of the Swiss 10-francbanknote and on numerous Swiss, German, and Russianpostage stamps. The asteroid 2002 Euler was named in hishonor.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Leonhard Euler, 1707-1783

• Leonhard Euler was a Swiss mathematicianwho made enormous contributions to a widerange of mathematics and physics includinganalytic geometry, trigonometry, geometry,calculus, number theory and graph theory.He published over 500 works.

• A statement attributed to Pierre-Simon Laplace expressesEuler’s influence on mathematics: “Read Euler, read Euler,he is the master of us all.”

• Euler was featured on the sixth series of the Swiss 10-francbanknote and on numerous Swiss, German, and Russianpostage stamps. The asteroid 2002 Euler was named in hishonor.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Euler’s Theorem

Euler’s Theorem A graph has an Euler circuit if and only if it isconnected and each vertex has an even degree.

Let’s see how to find an Euler circuit in a connected even graph:

• Start from any vertex, find a circuit C without repeated edges.• Remove the edges in C from the graph, and find Euler

circuits from each component of the remaining graph.• Combine C with the Euler circuits to get an Euler circuit in

the original graph.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Euler’s Theorem

Euler’s Theorem A graph has an Euler circuit if and only if it isconnected and each vertex has an even degree.

Let’s see how to find an Euler circuit in a connected even graph:

• Start from any vertex, find a circuit C without repeated edges.

• Remove the edges in C from the graph, and find Eulercircuits from each component of the remaining graph.

• Combine C with the Euler circuits to get an Euler circuit inthe original graph.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Euler’s Theorem

Euler’s Theorem A graph has an Euler circuit if and only if it isconnected and each vertex has an even degree.

Let’s see how to find an Euler circuit in a connected even graph:

• Start from any vertex, find a circuit C without repeated edges.• Remove the edges in C from the graph, and find Euler

circuits from each component of the remaining graph.

• Combine C with the Euler circuits to get an Euler circuit inthe original graph.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Euler’s Theorem

Euler’s Theorem A graph has an Euler circuit if and only if it isconnected and each vertex has an even degree.

Let’s see how to find an Euler circuit in a connected even graph:

• Start from any vertex, find a circuit C without repeated edges.• Remove the edges in C from the graph, and find Euler

circuits from each component of the remaining graph.• Combine C with the Euler circuits to get an Euler circuit in

the original graph.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Example: Euler circuit

• Find an Euler circuit in the following graph

1 2 4

5 6 7 8

3

• 12, 23, 34, 48, 87, 76, 65, 51• 16, 62, 27, 71• 52, 28, 83, 36, 64, 45• Euler circuit:

16, 62, 27, 71, 12, 23, 34, 48, 87, 76, 65, 52, 28, 83, 36, 64,45, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Example: Euler circuit

• Find an Euler circuit in the following graph

1 2 4

5 6 7 8

3

• 12, 23, 34, 48, 87, 76, 65, 51

• 16, 62, 27, 71• 52, 28, 83, 36, 64, 45• Euler circuit:

16, 62, 27, 71, 12, 23, 34, 48, 87, 76, 65, 52, 28, 83, 36, 64,45, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Example: Euler circuit

• Find an Euler circuit in the following graph

1 2 4

5 6 7 8

3

• 12, 23, 34, 48, 87, 76, 65, 51• 16, 62, 27, 71

• 52, 28, 83, 36, 64, 45• Euler circuit:

16, 62, 27, 71, 12, 23, 34, 48, 87, 76, 65, 52, 28, 83, 36, 64,45, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Example: Euler circuit

• Find an Euler circuit in the following graph

1 2 4

5 6 7 8

3

• 12, 23, 34, 48, 87, 76, 65, 51• 16, 62, 27, 71• 52, 28, 83, 36, 64, 45

• Euler circuit:16, 62, 27, 71, 12, 23, 34, 48, 87, 76, 65, 52, 28, 83, 36, 64,45, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Example: Euler circuit

• Find an Euler circuit in the following graph

1 2 4

5 6 7 8

3

• 12, 23, 34, 48, 87, 76, 65, 51• 16, 62, 27, 71• 52, 28, 83, 36, 64, 45• Euler circuit:

16, 62, 27, 71, 12, 23, 34, 48, 87, 76, 65, 52, 28, 83, 36, 64,45, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Eulerizing a graph

Ex: Does the following have an Euler circuit?

A

B C

D E

F

G

H

• How many edges do we need to repeat to cover all the edges?

—– we have to repeat AC and GH• Observe that the vertices without even degrees are A, C, G,

and H. Is this a coincidence?• Not really! If add edges AC and GH to the original graph, we

will get a graph with only even degrees, thus get an Eulercircuit.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Eulerizing a graph

Ex: Does the following have an Euler circuit?

A

B C

D E

F

G

H

• How many edges do we need to repeat to cover all the edges?

—– we have to repeat AC and GH• Observe that the vertices without even degrees are A, C, G,

and H. Is this a coincidence?• Not really! If add edges AC and GH to the original graph, we

will get a graph with only even degrees, thus get an Eulercircuit.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Eulerizing a graph

Ex: Does the following have an Euler circuit?

A

B C

D E

F

G

H

• How many edges do we need to repeat to cover all the edges?

—– we have to repeat AC and GH

• Observe that the vertices without even degrees are A, C, G,and H. Is this a coincidence?

• Not really! If add edges AC and GH to the original graph, wewill get a graph with only even degrees, thus get an Eulercircuit.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Eulerizing a graph

Ex: Does the following have an Euler circuit?

A

B C

D E

F

G

H

• How many edges do we need to repeat to cover all the edges?

—– we have to repeat AC and GH• Observe that the vertices without even degrees are A, C, G,

and H. Is this a coincidence?

• Not really! If add edges AC and GH to the original graph, wewill get a graph with only even degrees, thus get an Eulercircuit.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Eulerizing a graph

Ex: Does the following have an Euler circuit?

A

B C

D E

F

G

H

• How many edges do we need to repeat to cover all the edges?

—– we have to repeat AC and GH• Observe that the vertices without even degrees are A, C, G,

and H. Is this a coincidence?• Not really! If add edges AC and GH to the original graph, we

will get a graph with only even degrees, thus get an Eulercircuit.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even

• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Facts on graphs

• In order to do this, we should be able to pair up the verticeswith odd degrees.

• In other words, there are even number of odd-degree verticesin a graph. Is this always true?

• Yes! Here are two facts about graphs and the first one impliesthe second one

• The sum of all degrees is even• The number of odd-degree vertices is even

• Now we know the number of odd-degree vertices is alwayseven, so in order to create an Euler circuit in a non-Euleriangraph, we just need to identify the odd-degree vertices, pairthem up, add or repeat the corresponding edges.

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Ex. Find an Euler circuit in the following graph with fewestrepeated edges

1 2 43

9 10 11 12

5 6 7 8

• Identify the odd-degree vertices: 2, 3, 5, 8, 10, 12• Pair up the vertices so that the total distance between pairs is

smallest: (2, 3), (5, 8), and (10, 11) and add the edges• In the new graph, find an Euler circuit:

12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 85, 51

• Replace the added edges (red edges) with paths in theoriginal graph:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 87, 76, 65, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Ex. Find an Euler circuit in the following graph with fewestrepeated edges

1 2 43

9 10 11 12

5 6 7 8

• Identify the odd-degree vertices: 2, 3, 5, 8, 10, 12

• Pair up the vertices so that the total distance between pairs issmallest: (2, 3), (5, 8), and (10, 11) and add the edges

• In the new graph, find an Euler circuit:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 85, 51

• Replace the added edges (red edges) with paths in theoriginal graph:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 87, 76, 65, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Ex. Find an Euler circuit in the following graph with fewestrepeated edges

1 2 43

9 10 11 12

5 6 7 8

• Identify the odd-degree vertices: 2, 3, 5, 8, 10, 12• Pair up the vertices so that the total distance between pairs is

smallest: (2, 3), (5, 8), and (10, 11) and add the edges

• In the new graph, find an Euler circuit:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 85, 51

• Replace the added edges (red edges) with paths in theoriginal graph:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 87, 76, 65, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Ex. Find an Euler circuit in the following graph with fewestrepeated edges

1 2 43

9 10 11 12

5 6 7 8

• Identify the odd-degree vertices: 2, 3, 5, 8, 10, 12• Pair up the vertices so that the total distance between pairs is

smallest: (2, 3), (5, 8), and (10, 11) and add the edges• In the new graph, find an Euler circuit:

12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 85, 51

• Replace the added edges (red edges) with paths in theoriginal graph:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 87, 76, 65, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Ex. Find an Euler circuit in the following graph with fewestrepeated edges

1 2 43

9 10 11 12

5 6 7 8

• Identify the odd-degree vertices: 2, 3, 5, 8, 10, 12• Pair up the vertices so that the total distance between pairs is

smallest: (2, 3), (5, 8), and (10, 11) and add the edges• In the new graph, find an Euler circuit:

12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 85, 51

• Replace the added edges (red edges) with paths in theoriginal graph:12, 23, 34, 48, 87, 76, 65, 59, 9(10), (10)6, 62, 23, 37, 7(11),(11)(10), (10)(11), (11)(12), (12)8, 87, 76, 65, 51

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!

Chapter 1Urban Services

Chi-Kwong Li

Urban Services

Seven bridgeproblem

Solution ofEuler

Leonhard Euler

Euler’sTheorem

Eulerizing agraph

Facts on graphs

Furtherextension

Further Extension

• What if there are weights (lengths) associated with theedges? Can we find the circuit containing all edges withminimum weight? [Chinese postman problem.]

• What if there are directions associated with the graph, i.e.,we are dealing with a directed graph (digraph).

• Theorem A digraph has a directed Euler circuit if and only if. . . .

• Other problems?

The End!