Deck 5: The Mathematics of Getting Around: Euler Paths and Circuits

Question 1

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to
put his pencil on the drawing and trace each and every line once and only once, without lifting his
pencil. Is John able to complete this task? Explain.

Answer 1

No, as the drawing can be thought of
as a graph that has four vertices of
odd degree. Thus, neither an Euler
Path or an Euler Circuit can be found.

Question 2

Is it possible to construct a graph consisting of two vertices of even degree and three vertices of odd degree? Explain.

Answer 2

No. A graph will always have an
even number of odd degree vertices.

Question 3

Consider the graph above ; how many circuits consisting of three edges are there which
involve vertex A?

Answer 3

2, if you do not count mirror images
as being different. Otherwise, there
would be 4 if you count mirror
images as being different.

Question 4

Consider the graph with vertex set V = {A,,BC,X,Y,Z} and edge set E =
{ABA,,CAY,AX,BYC,C,CZ}. Which edges, if any, are bridges for the graph?

Question 5

For the graph shown below, which vertices are adjacent to vertex C?

Question 6

Consider the graph with vertex set V = {A,,BC,X,Y,Z} and edge set E =
{ABA,,CAY,AX,BYC,C,CZ}. Does this graph contain an Euler Path? Explain.

Question 7

Consider the graph with vertex set ={A, B, C, D, E, F} and edge set =
What is the degree of vertex E ?

Question 8

Consider the graph above ; state the vertex set.

Question 9

Consider the graph with vertex set ={A, B, C, D, E, F} and edge set = {AB, AB, AC, AE, AF, BD, BF, CD, CE, DF, EE, EF} . List all edges adjacent to edge D F .

Question 10

Consider the graph above ; state the degree of vertex A.

Question 11

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to put his pencil on the drawing and trace each and every line once and only once, without lifting his pencil. Is John able to complete this task? Explain.

Question 12

Tom is dancing at a local discothèque with his friends Matt, Bill, Lisa, and Allison. Tom dances with
Lisa once and Allison twice. Bill dances with Lisa two times but does not dance with Allison, and
Matt dances with Lisa and Allison one time each. Using people as vertices and dances as edges
construct a graph which models this situation.

Question 13

For the graph shown below, what is the edge set E?

Question 14

The graph below does not contain an Euler Circuit. How many edges are traveled a second time if an optimal Eulerization is found for the graph?

Question 15

How many edges are bridges in the following grapgh ?

Question 16

Consider the graph above , which edges, if any, are adjacent to edge BC?

Question 17

Your university has installed a new Meditation Garden which consists of several island gardens connected by bridges. A layout of the garden is shown below. You are enrolled in a Buddhism class and your instructor assigns you the following assignment.
Start at the Tranquility Station and walk through the Meditation Gardens. Every time that you cross a bridge you must recite a Buddhist verse. Continue walking through the gardens until you have crossed each and every bridge. End your journey at the Tranquility Station. What is the fewest number of verses that you would need to recite in order to complete the assignment?

Question 18

Explain why the graph in figure 2 is not an Eulerization of the graph in figure 1.

Question 19

Three distinguished faculty members meet four honors students at a local pub. Each faculty member
shakes hands with each student once and only once. Using people as vertices and handshakes as edges,
construct a graph which models this situation.

Question 20

For the graph shown below, state the edge list.

Question 21

A graph with two vertices of odd degree and three vertices of even degree will always contain at least one Euler Path. This statement is

A) always true.
B) always false.
C) true only if the graph is connected.
D) false only if the graph is connected.
E) None of the above.

Question 22

Your college campus has designed an underground tumnel system to connect five of the main buildings on campus. A map of the underground system is shown below. Suppose that you start in Meconi Hall, end in The Glasgow Center, and plan to travel through each and every tunnel, what is the fewest number of tunnels that you would need to travel through more than once in order to accomplish this task?

A) 0
B) 1
C) 2
D) 3
E) None of the above.

Question 23

Consider the graph with vertex set ={G, H, O, S, T} and edge set = {G H, G S, G S, G T, H O, O T, S T} . Which edges, if any, are bridges for the graph?

A) G H
B) G H and G T
C) O T
D) All of the edges.
E) None of the above.

Question 24

The graph below does not contain an Euler Circuit. If an optimal Eulerization is found for the graph, then how many edges will be traveled over a second time?

A) 1
B) 2
C) 3
D) 4
E) None of the above.

Question 25

Consider the graph above ; which edges are adjacent to edge AD ?

A) AB, AC, AE, BD, CD, ED
B) AB, BB, BD, CD, ED
C) AB, BD, CD
D) AC, A E, BD, BE, ED
E) None of the above.

Question 26

Consider the graph with vertex set and edge set
What is the degree of vertex H ?

A) 0
B) 1
C) 2
D) 3
E) None of the above.

Question 27

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to put his pencil on the drawing and trace each and every line once and only once, without lifting his pencil. Is John able to complete this task?

A) Yes
B) No

Question 28

How many edges are bridges in the following graph?

A) 0
B) 1
C) 2
D) 3
E) None of the above.

Question 29

Consider the graph above ; how many paths are there which start at vertex A and end at vertex G?

A) 7
B) 8
C) 9
D) 10
E) None of the above.

Question 30

Consider the graph above ; what is the vertex set?

A) A, B, C, D, E, F, G
B) A, A, B, B, B, C, C, C, D, D, D, E, E, E, F, F, G, G
C) A B, A D, B C, B D, C D, C E, E F, E G, F G
D) A B, A D, B C, B D, C D, E F, F G
E) None of the above.

Question 31

Consider the graph with vertex set ={G, H, O, S, T} and edge set = {G H, G S, G S, G T, H O, O T, S T} . This graph

A) contains at least one Euler Path.
B) contain at least one Euler Circuit.
C) is connected but does not contain either an Euler Path or an Euler Circuit.
D) is not connected.
E) None of the above.

Question 32

Two math professors and three engineering professors meet at a local pub. Each professor shakes hands with every other professor once and only once. Which of the graphs below models this situation?

A)
B)
C)
D)
E) None of the above.

Question 33

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to
put his pencil on the drawing and trace each and every line once and only once, without lifting his
pencil. Is John able to complete this task?

(a) Yes
(b) No

Question 34

For the graph shown below, what is the degree of vertex D?

A) 2
B) 3
C) 4
D) 5
E) N one of the above.

Question 35

Three countries align with each other and form the Axis of Evil while four other countries align with each other and form the Axis of Good. Which graph models this situation?

A)
B)
C)
D)
E) None of the above.

Question 36

The graph shown below

A) contains at least one Euler Path.
B) contain at least one Euler Circuit.
C) is connected but does not contain either an Euler Path or an Euler Circuit.
D) is not connected.
E) None of the above.

Question 37

Consider the graph with vertex set and edge set Which edges, if any, are adjacent to edge GT ?

A) H, S, T
B) G H, G S, O T, S T
C) A O
D) G S, H O, S T
E) None of the above.