Euler Graphs

Section 6.2

6.2 Euler Graphs 2

Circuit? Path? Non- traversable?

A

D

E

C

B

A

D

E

C

B

A

D

E

C

B

End at A

End at B

Start at A

Start at A Miss an

edge

Start at A

6.2 Euler Graphs 3

Circuit? Path? Non-traversable?

A

E

C

B

G

O

I

F

K

P

N

H

J

M

D

16

Vertices

28 Edges

L

6.2 Euler Graphs 4

Stump the Prof

6.2 Euler Graphs 5

Graph Vertices Edges Type

?

6.2 Euler Graphs 6

Conclusion

Therefore the type of graph is not determined by

• .

• .

So, what is it determined by?

6.2 Euler Graphs 7

Does the graph have a Euler circuit? path? or neither. What is the degree of each vertex? Click “yes” if you see a pattern.1. Yes2. No

6.2 Euler Graphs 8

Make a hypothesis based on your work

Verify by filling in last column on next slide

6.2 Euler Graphs 9

Non-traversab

le

Circuit

Path

Graph Vertices Edges Type

?

5

5

5

9

9

9

6.2 Euler Graphs 10

Euler’s Theorem

Let N = the number of vertices in a graph

• If N = the graph has an Euler Circuit (EC)

• If N = the graph has an Euler Path (EP) (Must start at an odd vertex)

• If N = (or more), the graph is Non-Traversable (NT)

6.2 Euler Graphs 11

Solution to the Konigsberg Bridge Problem

6.2 Euler Graphs 12

Can you draw this figure without taking your pencil from the paper and without retracing any line?

1. Yes, and I can start at any vertex

2. Yes, but only if I start at certain vertices

3. No, it can’t be done no matter where I start

6.2 Euler Graphs 13

I see how Euler’s Theorem applies to this problem

1. Absolutely

2. Sort of

3. Not a clue

6.2 Euler Graphs 14

Snow Plowing

1. Circuit

2. Path

3. Non-traversable

Union City

Dover

Paterson

Morristown

Clifton

Hackensack

Passaic

East Orange

6.2 Euler Graphs 15

“Eulerizing”

1. 1

2. 2

3. 3

4. 4

What is the minimum number of roads that can be removed so that this graph will have an Euler Circuit?

Union City

Dover

Paterson

Morristown

Clifton

Hackensack

Passaic

East Orange

6.2 Euler Graphs 16

Security Guard Animation

6.2 Euler Graphs 17

Can a security guard make an Euler Circuit starting at the parking lot

Yes N

o

50%50%

1. Yes

2. No

Employee Parking Lot

N

C

F

D EBA

M

K

J

I

L

HG

6.2 Euler Graphs 18

What is the minimum number of doors that can be removed so that this floor plan will have an Euler Circuit?

0% 0%0%

100%

1. 1

2. 2

3. 3

4. 4

Employee Parking

Lot

N

C

F

D EBA

MK

J

I

L

HG

6.2 Euler Graphs 19

Can you cross each of the borders between pairs of neighboring New England states once and only once and return to the state from which you started?0% 0%

100%

1. Yes

2. No

3. Sometimes

6.2 Euler Graphs 20

Here is Euler’s graph of Konigsberg. Can you start at some vertex and cross every edge twice and only twice?

Yes N

o

Can

’t te

ll

100%

0%0%

1. Yes

2. No

3. Can’t tell

A

C

D

B

6.2 Euler Graphs 21

A final question

Why didn’t Euler worry about the cases of 1, 3, 5, … odd vertices?

Draw a graph with one odd vertex.

6.2 Euler Graphs 22

Hypothesis:

Counting vertices, edges, degrees applet

6.2 Euler Graphs 23

If you could draw a graph with exactly one odd vertex, the sum of all the degrees would be .

Odd

Even

Even

Even

Even

Even

Even

6.2 Euler Graphs 24

Theorem: The sum of the degrees of

all vertices of a graph is 2 * (

).

Corollary: The sum of the degrees of all

vertices of a graph is always

.

6.2 Euler Graphs 25

Can a graph have nine edges of which

4 have degree 2, three have degree 3

and two have degree 4?

1. Yes

2. No

6.2 Euler Graphs 26

Can a graph have nine edges of which

4 have degree 3, three have degree 4

and two have degree 2?

1. Yes

2. No

6.2 Euler Graphs 27

A graph has 6 vertices of degree 3 and 5 vertices of degree 4. How many edges does the graph have?

1. 11

2. 19

3. 38

6.2 Euler Graphs 28

End of 6.2

6.2 Euler Graphs 29

N

C

F

D EBA

M

KJ

I

L

HG

Drats!I’m Stuck

Employee Parking

LotStart

6.2 Euler Graphs 30

What is the security man trying to do?

6.2 Euler Graphs 31

N

C

F

D EBA

M

KJ

I

L

HG

Darn!I got all the doors, but…

Start

Employee Parking

Lot

6.2 Euler Graphs 32

In floor plans the vertices are

The

room

s

The

doors

50%50%

1. The rooms

2. The doors

6.2 Euler Graphs 33

N

C

F

D EBA

M

KJ

I

L

HG

All Clear!!

Employee Parking

LotStart

6.2 Euler Graphs 34

Draw a graph with

• 4 vertices (all odd) and 6 edges

• 4 vertices (all odd) and 3 edges

6.2 Euler Graphs 35

Draw a graph with

• 4 vertices (all even) and 5 edges (loops are edges)

• 5 vertices (3 even) and 8 edges

6.2 Euler Graphs 36

But

6.2 Euler Graphs 37

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