Aim: Graph Theory – Euler Paths & Circuits Course: Math Literacy Do Now: Aim: What is an Euler Path and Circuit? Represent the following with a graph

Aim: Graph Theory – Euler Paths & Circuits

Course: Math Literacy

Do Now:

Aim: What is an Euler Path and Circuit?

Represent the following with a graph

C D

BA

A B

C D



Euler Path

Euler Path – a path that travels through every edge of a graph once and only once. Each edge must be traveled and no edge can be retraced.

A

B C

F

D

GE

1

A, B, E, D, B, C, E, D,

2

34

5

6

78

9

GF,



A

B C

F

D

GE

Euler Path

Euler Circuit – a circuit that travels through every edge of a graph once and only once, and must begin and end at the same vertex.

1 A, B, E, D,

B, C, E, D,

2

34

5

6

78

9G,

F,

A10

Every Euler circuit is an Euler pathNot every Euler path is an Euler circuitSome graphs have no Euler pathsOther graphs have several Euler pathsSome graphs with Euler paths

have no Euler circuits



Euler Theorem

Euler’s Theorem

The following statements are true for connected graphs:

1. If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other.

2. If a graph has no odd vertices (all even vertices), it has at least one Euler circuit. An Euler circuit can start and end at any vertex.

3. If a graph has more than two odd vertices, then it has no Euler paths and no Euler circuits.



Model Problem

Explain why the graph has at least one Euler path.

D

CB

E

A number of edges at each vertex:

A:B:C:

D:E:

2 344

3

Two odd vertices


degree 4degree 4

degree 3 degree 3

degree 2



Model Problem

Find a Euler path.

D

CB

E

A

3

25

4

6

8

7

1

D, C, B, C, A, B, D, EE,



Model Problem

Explain why the graph has a least one Euler circuit.

A B

C D

HG

I J

E F

degree 2

degree 2

degree 2degree 2

degree 4 degree 4

degree 4 degree 4

2. If a graph has no odd vertices (all even vertices), it has at least one Euler circuit. An Euler circuit can start and end at any vertex.

no odd vertices



Model Problem

Find an Euler circuit.

H, G, E, G, I, J, H, D,C,

Euler Circuit – a circuit that travels through every edge of a graph once and only once, and must begin and end at the same vertex.

A B

C D

HG

I J

E F2

34

6

5

8

7

109

12

11

14

13

1

C, A, B, D, F, H



Graph Theory’s Beginnings

In the early 18th century, the Pregel River in a city called Konigsberg, surround an island before splitting into two. Seven bridges crossed the river and connected four different land area. Many citizens wished to take a stroll that would lead them across each bridge and return them to the starting point without traversing the same bridge twice. Possible?

They couldn’t do it.Euler proved that it was not possible.



The Theorem at WorkR

L

AB

R

L

AB

degree 3

degree 5

degree 3

3. If a graph has more than two odd vertices, then it has no Euler paths and not Euler circuits.



Model Problem

A, B, C, and D represent rooms. The outside of the house is labeled E. The openings represent doors.

a) Is it possible to find a path that uses each door exactly once?

C

A

E

B

D

degree 6

C

A

E

B

D

look for a Euler path or circuit

degree 3

exactly 2 odd vertices




Model Problem

A, B, C, and D represent rooms. The outside of the house is labeled E. The openings represent doors.

b) If possible, find such a path.

C

A

E

B

D

C

A

E

B

D

12

3

4

5

6

7

8

109

B, E, A, D, C, A, E, C,B, E, D

start at one of the odd vertices



Fluery’s Algorithm

If Euler’s Theorem indicates the existence of an Euler path or Euler circuit, one can be found using the following procedure.

1. If the graph has exactly two odd vertices, chose one of the two odd vertices as the starting point. If the graph has no odd vertices, choose any vertex as the starting point.

2. Number edges as you trace through the graph according to the following rules:

After you traveled over an edge, change it to a dashed line.

When faced with a choice of edges to trace, choose an edge that is not a bridge (an edge, which, if removed from a connected graph would leave behind a disconnected graph). Travel over an edge that is a bridge only if there is no alternative.



Model Problem

The graph has at least one Euler circuit. Find it using Fleury’s Algorithm.

EF

D C

B A

no odd vertices, begin at any vertex.

1

2

34

5

6

7

8

9

Documents

Aim: Graph Theory – Euler Paths & Circuits Course: Math Literacy Do Now: Aim: What is an Euler Path and Circuit? Represent the following with a graph