homework collection thursday 3/29 read pages 160 – 174 page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 –...
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![Page 1: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/1.jpg)
Homework collection Thursday 3/29
• Read Pages 160 – 174
• Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20
![Page 2: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/2.jpg)
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.
![Page 3: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/3.jpg)
1. Try to trace the following without lifting up your pencil or retracing any lines, so that:
You start and finish at different places
![Page 4: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/4.jpg)
1. Try to trace the following without lifting up your pencil or retracing any lines, so that:
You finish at the starting place
![Page 5: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/5.jpg)
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.
![Page 6: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/6.jpg)
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
![Page 7: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/7.jpg)
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.
![Page 8: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/8.jpg)
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).
![Page 9: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/9.jpg)
Objective 5:Proving Theorems
![Page 10: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/10.jpg)
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.
![Page 11: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/11.jpg)
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.
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# 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
![Page 13: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/13.jpg)
Do the following have Euler Paths, Euler Circuits, or neither?
![Page 14: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/14.jpg)
• Is it possible to take a walk through town crossing each of the bridges once and only once?
![Page 15: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/15.jpg)
![Page 16: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/16.jpg)
A
BC
D
E
FG H
IJ
K
Does the following have an Euler path, Euler circuit, both, or neither?
L
![Page 17: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/17.jpg)
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
![Page 18: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/18.jpg)
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!)
![Page 19: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/19.jpg)
Do the following have Euler Paths, Euler Circuits, or neither?
![Page 20: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/20.jpg)
• 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)
![Page 21: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/21.jpg)
• 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
![Page 22: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/22.jpg)
A B C D
J I H G
E
FK
L
FIGURE 5-22 Page 180
![Page 23: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/23.jpg)
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
![Page 24: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/24.jpg)
• The process of turning odd vertices into even vertices by adding “duplicate” edges in strategic areas is called eulerization (page 179)
![Page 25: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/25.jpg)
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
![Page 26: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/26.jpg)
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
![Page 27: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/27.jpg)
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
![Page 28: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/28.jpg)
FIGURE 5-24 page 181 - SEMIEULERIZATION
A B C D E
M L K J I
N
O
P F
G
H
(a)
![Page 29: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/29.jpg)
A B C D E
M L K J I
N
O
P F
G
H
(a)
FIGURE 5-24 page 181 - SEMIEULERIZATION
![Page 30: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/30.jpg)
• 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
![Page 31: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/31.jpg)
• 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
![Page 32: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/32.jpg)
EXAMPLE 13:(page 189)
![Page 33: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/33.jpg)
![Page 34: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/34.jpg)
S
![Page 35: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/35.jpg)
S
![Page 36: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/36.jpg)
S
![Page 37: Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d825503460f94a67090/html5/thumbnails/37.jpg)
Homework
• Read Pages 175 – 184
• Page 188: 21 – 26, 29 – 36, 41 – 44, 47 – 49, 51, 52, 59, 62, 63