homework collection thursday 3/29 read pages 160 – 174 page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 –...

Homework collection Thursday 3/29

• Read Pages 160 – 174

• Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20

Euler paths (page 169). An Euler path is a path that travels through (exactly once) every edge of a graph.

Euler circuits (page 169). An Euler circuit is a circuit that travels through (exactly once) every edge of a graph.

1. Try to trace the following without lifting up your pencil or retracing any lines, so that:

You start and finish at different places

1. Try to trace the following without lifting up your pencil or retracing any lines, so that:

You finish at the starting place

Euler paths (page 169). An Euler path is a path that travels through (exactly once) every edge of a graph.

Euler circuits (page 169). An Euler circuit is a circuit that travels through (exactly once) every edge of a graph.

A D

R

L (a)

A D

R

L (b)

Euler paths (page 169). An Euler path is a path that travels through (exactly once) every edge of a graph.

Euler circuits (page 169). An Euler circuit is a circuit that travels through (exactly once) every edge of a graph.

Has no Euler paths Has several Euler paths

Neither has an Euler circuit

A B

CDE

F

G

(a)

A B

CDE

F

G

(b)

A B

CDE

F

G

(c)

Connected graphs. A graph is connected if any two of its vertices can be joined by a path.

Euler’s Theorem 1 (page 172)(a) If a graph has any vertices of odd degree, then it cannot have an Euler circuit.(b) If a graph is connected and every vertex has an even

degree, then it has at least one Euler circuit (and usually more).

Objective 5:Proving Theorems

Euler’s Theorem 2 (Page 172)(a) If a graph has more than two vertices of odd degree, then

it cannot have an Euler path.(b) If a graph is connected and has just two vertices of odd

degree, then it has at least one Euler path (and usually more). Any such path must start at one of the odd degree vertices and end at the other.

Euler’s Theorem 3 (page 174)

(a) The sum of the degrees of all vertices of a graph equals twice the number of edges (and therefore must be an even number)

(b) The number of vertices of odd degree must be even.

# of odd verticesConclusion (for connected graph)

0

Euler circuit exists (therefore Euler path also exists)

2Euler path exists but no Euler circuit

More than 2No Euler path or circuit

Do the following have Euler Paths, Euler Circuits, or neither?

• Is it possible to take a walk through town crossing each of the bridges once and only once?

A

BC

D

E

FG H

IJ

K

Does the following have an Euler path, Euler circuit, both, or neither?

L

A

BC

D

E

FG H

IJ

K

12

34

Does the following have an Euler path, Euler circuit, both, or neither?

Bridges. If a graph becomes disconnected by the removal of an edge, then that edge is called a bridge.

L

Fluery’s algorithm for finding an Euler Circuit

(page 176)

1. First make sure that the graph is connected and all the vertices have an even degree.

2. Start at any vertex.3. Travel through an edge if

(a) it is not a bridge for the untraveled part, or(b) there is no other alternative.

4. Label the edges in the order in which you travel them.5. When you can’t travel any more, stop. (You are done!)

Do the following have Euler Paths, Euler Circuits, or neither?

• A garbage man intends to collect garbage from both sides of the street as he drives through city blocks. What is the minimum number of blocks that he will have to drive? (assume he can start and finish at different points)

• A garbage man intends to collect garbage from both sides of the street as he drives through city blocks. What is the minimum number of blocks that he will have to drive? (assume he can start and finish at different points)

S

A B C D

J I H G

E

FK

L

FIGURE 5-22 Page 180

A B C D

J I H G

E

FK

L

A B C D

J I H G

E

FK

L

A B C D

J I H G

E

FK

L

5

48

1

3

7

9

11

12

141516

18 19 20

22

2325

26

28

2,10

13,21

6,24

17,27

A B C D

J I H G

E

FK

L

10

21

24

27 48

1 23

67

9

11

12

13

141516

17

18 19 20

22

2325

26

28

5

FIGURE 5-22 Page 180

• The process of turning odd vertices into even vertices by adding “duplicate” edges in strategic areas is called eulerization (page 179)

A B C D E

M L K J I

N

O

P F

G

H

A B C D E

M L K J I

N

O

P F

G

H

A B C D E

M L K J I

N

O

P F

G

H

(b)

(c) (d)

A B C D E

M L K J I

N

O

P F

G

H

(a)FIGURE 5-22 Page 180

A B C D E

M L K J I

N

O

P F

G

H

A B C D E

M L K J I

N

O

P F

G

H

A B C D E

M L K J I

N

O

P F

G

H

(b)

(c) (d)

A B C D E

M L K J I

N

O

P F

G

H

(a)FIGURE 5-22 Page 180

Not an Euler

izatio

n

Not an optim

al Euler

izatio

n

A B C D E

M L K J I

N

O

P F

G

H1,2

12,13

5

6

7

17

10

11

14 1615

18,21

19,20

12,13

3,4

8,9 35

25 24 23

223733

34 36

27 32 38 44,45

28

29 30

48 47 46

43

42

31 39

40,41

26

FIGURE 5-24 page 181 - SEMIEULERIZATION

A B C D E

M L K J I

N

O

P F

G

H

(a)

A B C D E

M L K J I

N

O

P F

G

H

(a)

FIGURE 5-24 page 181 - SEMIEULERIZATION

• A garbage man intends to collect garbage from both sides of the street as he drives through city blocks. What is the minimum number of blocks that he will have to drive? (assume he can start and finish at different points)

S

• A garbage man intends to collect garbage from both sides of the street as he drives through city blocks. What is the minimum number of blocks that he will have to drive? (assume he can start and finish at different points)

• Draw a route for the garbage man so that he travels the minimum number of blocks

S

EXAMPLE 13:(page 189)

S

S

S

Homework

• Read Pages 175 – 184

• Page 188: 21 – 26, 29 – 36, 41 – 44, 47 – 49, 51, 52, 59, 62, 63