Math 1 Unit 4 Answer Key By: Michelle Tibbetts and Jennalee Tracy 1. Determine which graph is an Euler circuit or an Euler path. If so, indicate the route by listing the order of the vertices. If not, explain why. a.

A

B E

C

D

This is an Euler path. You must start and end on either A or B ( one of the odd vertices) A-C-D-B-E-A-B B-D-C-A-E-B-A b.

A

B

D

E I

C

F

H

G J

This is an Euler circuit. All vertices are an even degree. A-B-C-G-H-J-I-E-F-B-E-D-A

2. Eulerize the following graph, using a minimum of repeated edges.

B

The least amount of repeated edges is 4. 3.

a. How do you determine an Euler circuit? It is a route through a connected graph where: 1) Each edge of the graph is traced exactly once. 2) The route starts and ends at the same vertex. 3) Each vertex must have an even degree. b. How do you determine an Euler path? An Euler Path is a route through a vertex edge model where: 1) Each edge is traced only once. 2) You begin and end at an odd vertex 3) Only two vertices may be odd.

4. What are the advantages in using matrices when looking at vertex-edge graphs? Matrices are another way to help organize a graph and show the number of edges and vertices. a. How do you determine row sums? - Add all the numbers in each row to find the row sum. b. What are the row sums of this matrix? A A0 B2 C2 D0 E0

5.

B 2 0 0 1 1

C 2 0 0 1 1

D 0 1 1 0 2

E 0 1 1 2 0

4 4 4 4 4

Here is a list of tasks that need to be completed for Susie’s Dance Recital.

Task Pay the bill Arrive for opening night Sign up for classes Attend dress rehearsal Learn the routines Order costumes

B N C R L O

Task Time 1 hour 4 hours 1 hour 5 hours 15 hours 2 hours

Immediate Prerequisite C R -L None O B B

a. Using the prerequisites in question 5, create a digraph.

L 15 S

C 1

N 4

B 1

F

O 2

R 5

b. What is the EFT for the entire project? 21 hours would be the EFT. c. What is the critical path? SCBLNF

7. Color the following map with the least amount of colors so that no two borders share the same color.

The least amount of colors needed is three.

A

B E

C

D

This is an Euler path. You must start and end on either A or B ( one of the odd vertices) A-C-D-B-E-A-B B-D-C-A-E-B-A b.

A

B

D

E I

C

F

H

G J

This is an Euler circuit. All vertices are an even degree. A-B-C-G-H-J-I-E-F-B-E-D-A

2. Eulerize the following graph, using a minimum of repeated edges.

B

The least amount of repeated edges is 4. 3.

a. How do you determine an Euler circuit? It is a route through a connected graph where: 1) Each edge of the graph is traced exactly once. 2) The route starts and ends at the same vertex. 3) Each vertex must have an even degree. b. How do you determine an Euler path? An Euler Path is a route through a vertex edge model where: 1) Each edge is traced only once. 2) You begin and end at an odd vertex 3) Only two vertices may be odd.

4. What are the advantages in using matrices when looking at vertex-edge graphs? Matrices are another way to help organize a graph and show the number of edges and vertices. a. How do you determine row sums? - Add all the numbers in each row to find the row sum. b. What are the row sums of this matrix? A A0 B2 C2 D0 E0

5.

B 2 0 0 1 1

C 2 0 0 1 1

D 0 1 1 0 2

E 0 1 1 2 0

4 4 4 4 4

Here is a list of tasks that need to be completed for Susie’s Dance Recital.

Task Pay the bill Arrive for opening night Sign up for classes Attend dress rehearsal Learn the routines Order costumes

B N C R L O

Task Time 1 hour 4 hours 1 hour 5 hours 15 hours 2 hours

Immediate Prerequisite C R -L None O B B

a. Using the prerequisites in question 5, create a digraph.

L 15 S

C 1

N 4

B 1

F

O 2

R 5

b. What is the EFT for the entire project? 21 hours would be the EFT. c. What is the critical path? SCBLNF

7. Color the following map with the least amount of colors so that no two borders share the same color.

The least amount of colors needed is three.