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...
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![Page 1: 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 Miss an edge](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d5f5503460f94a40038/html5/thumbnails/1.jpg)
Euler Graphs
Section 6.2
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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
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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
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6.2 Euler Graphs 4
Stump the Prof
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6.2 Euler Graphs 5
Graph Vertices Edges Type
?
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6.2 Euler Graphs 6
Conclusion
Therefore the type of graph is not determined by
• .
• .
So, what is it determined by?
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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
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6.2 Euler Graphs 8
Make a hypothesis based on your work
Verify by filling in last column on next slide
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6.2 Euler Graphs 9
Non-traversab
le
Circuit
Path
Graph Vertices Edges Type
?
5
5
5
9
9
9
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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)
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6.2 Euler Graphs 11
Solution to the Konigsberg Bridge Problem
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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
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6.2 Euler Graphs 13
I see how Euler’s Theorem applies to this problem
1. Absolutely
2. Sort of
3. Not a clue
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6.2 Euler Graphs 14
Snow Plowing
1. Circuit
2. Path
3. Non-traversable
Union City
Dover
Paterson
Morristown
Clifton
Hackensack
Passaic
East Orange
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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
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6.2 Euler Graphs 16
Security Guard Animation
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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
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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
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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
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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
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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.
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6.2 Euler Graphs 22
Hypothesis:
Counting vertices, edges, degrees applet
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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
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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
.
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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
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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
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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
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6.2 Euler Graphs 28
End of 6.2
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6.2 Euler Graphs 29
N
C
F
D EBA
M
KJ
I
L
HG
Drats!I’m Stuck
Employee Parking
LotStart
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6.2 Euler Graphs 30
What is the security man trying to do?
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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
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6.2 Euler Graphs 32
In floor plans the vertices are
The
room
s
The
doors
50%50%
1. The rooms
2. The doors
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6.2 Euler Graphs 33
N
C
F
D EBA
M
KJ
I
L
HG
All Clear!!
Employee Parking
LotStart
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6.2 Euler Graphs 34
Draw a graph with
• 4 vertices (all odd) and 6 edges
• 4 vertices (all odd) and 3 edges
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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
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6.2 Euler Graphs 36
But
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6.2 Euler Graphs 37
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