Section 6 2 Euler Graphs Circuit Path Start

Section 6. 2 Euler Graphs

Circuit? Path? Start at A End at A Start at A End at B A Nontraversable? Start at A Miss an edge A A B B B D D C C E E 6. 2 Euler Graphs D C E 2

Circuit? Path? Non-traversable? 16 Vertices 28 Edges A G B D F H C I E J K L M O N P 6. 2 Euler Graphs 3

Stump the Prof 6. 2 Euler Graphs 4

Graph Vertices Edges 6. 2 Euler Graphs Type ? 5

Conclusion Therefore the type of graph is not determined by • . So, what is it determined by? 6. 2 Euler Graphs 6

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. Yes 2. No 6. 2 Euler Graphs 7

Make a hypothesis based on your work Verify by filling in last column on next slide 6. 2 Euler Graphs 8

Graph Vertices Edges Type 5 9 Path 5 9 Nontraversable 5 9 Circuit 6. 2 Euler Graphs ? 9

Euler’s Theorem Let N = the number of • If N = vertices in a graph 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 10

Solution to the Konigsberg Bridge Problem 6. 2 Euler Graphs 11

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 12

I see how Euler’s Theorem applies to this problem 1. Absolutely 2. Sort of 3. Not a clue 6. 2 Euler Graphs 13

Snow Plowing Paterson Hackensack Passaic Dover Clifton Union City Morristown East Orange 1. Circuit 2. Path 3. Non-traversable 6. 2 Euler Graphs 14

“Eulerizing” What is the minimum number of roads that can be removed so that this graph will have an Euler Paterson Circuit? Hackensack Dover Clifton Morristown 1. 1 2. 2 3. 3 4. 4 East Orange 6. 2 Euler Graphs Passaic Union City 15

Security Guard Animation 6. 2 Euler Graphs 16

A B C F G H D E Can a security guard make an Euler Circuit starting at the parking lot I J K M L N Employee Parking Lot 1. Yes 2. No 6. 2 Euler Graphs 17

A B C F G H D E What is the minimum number of doors that can be removed so that this floor plan will have an Euler Circuit? I J K M L N Employee Parking Lot 1. 1 2. 2 3. 3 4. 4 6. 2 Euler Graphs 18

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? 1. Yes 2. No 3. Sometimes 6. 2 Euler Graphs 19

A B C Here is Euler’s graph of Konigsberg. Can you start at some vertex and cross every edge twice and only twice? D 1. Yes 2. No 3. Can’t tell 6. 2 Euler Graphs 20

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 21

Counting vertices, edges, degrees applet Hypothesis: 6. 2 Euler Graphs 22

If you could draw a graph with exactly one odd vertex, the sum of all the degrees would be . Odd Even Even 6. 2 Euler Graphs 23

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 24

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 25

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 26

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 27

End of 6. 2 Euler Graphs 28

Drats! I’m Stuck A B C F G H D E I J L K M N Employee Parking Lot Start 6. 2 Euler Graphs 29

What is the security man trying to do? 6. 2 Euler Graphs 30

Darn! I got all the doors, but… A B C F G H D E I J K M L N Employee Parking Lot Start 6. 2 Euler Graphs 31

In floor plans the vertices are 1. The rooms 2. The doors 6. 2 Euler Graphs 32

All Clear!! A B C F G H D E I J L K M N Employee Parking Lot Start 6. 2 Euler Graphs 33

Draw a graph with • 4 vertices (all odd) and 6 edges • 4 vertices (all odd) and 3 edges 6. 2 Euler Graphs 34

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 35

But 6. 2 Euler Graphs 36

Meta - Material 6. 2 Euler Graphs 37