Question 1

Suppose that a graph containing four vertices A, B, C, and D has an Euler circuit starting from A. If the degree of the vertex B is 4, then how many times will the Euler circuit pass the vertex B?
A)1
B)2
C)3
D)4

Accepted Answer

Since the graph has an Euler circuit starting from A, all vertices must have even degree. If the degree of B is 4, then it must be visited twice during the circuit, once as an entry point and once as an exit point. Therefore, the Euler circuit will pass the vertex B twice.

Question 2

A circuit or cycle in a graph is a path that begins and ends at the same vertex.

Accepted Answer

A circuit or cycle in a graph is indeed a path that begins and ends at the same vertex.

Question 3

What is the degree of each vertex in the figure below? 
A)3
B)4
C)5
D)6

Accepted Answer

Each vertex is connected to 5 other vertices, so the degree of each vertex is 5.

Question 4

When graphs are represented pictorially using dots and segments, the dots are called ___________.
A)points
B)vertices
C)stops
D)edges

Accepted Answer

In graph theory, when graphs are represented pictorially, the dots are referred to as "vertices." These vertices (or nodes) are connected by lines or curves called "edges."

Question 5

How many Euler circuits starting from the vertex B are possible? 
A)4
B)6
C)8
D)12

Accepted Answer

The answer of How many Euler circuits starting from the...

Question 6

Find an Euler circuit for the figure below. 
A)ABDCA
B)ABDACDA
C)ADBACD
D)No Euler circuit exists.

Accepted Answer

The answer of Find an Euler circuit for the figure...

Question 7

According to Euler's theorem, the figure below has an Euler circuit.

Accepted Answer

Euler's theorem states that an undirected graph has an Euler circuit if and only if every vertex has an even degree. In the figure below, each vertex has degree 4, which is an even number. Therefore, the figure has an Euler circuit.

![image.png](attachment:image.png)

Question 8

The _______________ of a vertex is the number of edges that touch that vertex.
A)magnitude
B)amplitude
C)period
D)degree

Accepted Answer

The degree of a vertex is a graph theory concept that refers to the number of edges that are connected to a vertex. This is a fundamental concept in graph theory, and knowing the degree of a vertex can help determine the behavior and properties of a graph, such as determining whether the graph is connected or disconnected, or whether it is bipartite or not.

Question 9

If a circuit traverses each edge of the graph exactly once, it is called a(n) _____________ circuit.
A)traveling
B)single
C)Hamilton
D)Euler

Accepted Answer

An Euler circuit in a graph is a circuit that uses every edge of the graph exactly once.

Question 10

Suppose the edges of a certain graph represent phone lines that must be maintained, and the vertices represent junctions. A worker maintaining the lines would like to find an Euler circuit for this graph to increase ___________.
A)payment
B)mileage
C)efficiency
D)workload

Accepted Answer

Finding an Euler circuit means that the worker can visit every edge of the graph exactly once, which would save time and effort compared to visiting edges multiple times or missing edges entirely. This would therefore increase efficiency. Payment, mileage, and workload would not be directly affected by finding an Euler circuit.

Question 11

The computers in an office area are connected as follows:
A)The computer in Office A is connected to all computers.
B)The computer in Office B is connected to Offices A and D.
C)The computer in Office C is connected to Offices A and D.
D)The computer in Office D is connected to Offices A, B, and C.

Accepted Answer

The answer of The computers in an office area are...

Question 12

What is the degree of vertex F? 
A)2
B)3
C)4
D)5

Accepted Answer

The answer of What is the degree of vertex F?...

Question 13

Find an Euler circuit for the figure below. 
A)ABCDEFA
B)ABCDFEA
C)AFDECBA
D)No Euler circuit exists.

Accepted Answer

The answer of Find an Euler circuit for the figure...

Question 14

Find an Euler circuit for the figure below. 
A)AEFDCBA
B)EABCDFE
C)DFEABCD
D)No Euler circuit exists.

Accepted Answer

The answer of Find an Euler circuit for the figure...

Question 15

What is the degree of each vertex in the figure below? 
A)2
B)3
C)4
D)5

Accepted Answer

Each vertex of the figure is the meeting point of 4 line segments, hence the degree of each vertex is 4. Therefore, the correct choice is C.

Question 16

Which of the following vertices have degree 2? 
A)A and B
B)B and C
C)C and D
D)A and D

Accepted Answer

The answer of Which of the following vertices have degree...

Question 17

According to Euler's theorem, a connected graph has an Euler circuit precisely when every vertex has a(n) ___________ degree.
A)nonzero
B)positive
C)even
D)odd

Accepted Answer

Euler's theorem states that a connected graph has an Euler circuit if and only if every vertex has an even degree. Therefore, the correct choice is C, which corresponds to even degrees.

Question 18

Which of the following vertices have degree 2? 
A)A and B
B)C and D
C)B and E
D)D and F

Accepted Answer

The answer of Which of the following vertices have degree...

Question 19

According to Euler's theorem, the figure below has an Euler circuit.

Accepted Answer

Euler's theorem states that a graph has an Euler circuit if and only if every vertex has even degree. In the given figure, vertex B has odd degree (degree 3), therefore, there cannot be an Euler circuit.

Question 20

Find an Euler circuit for the figure below. 
A)ABDCA
B)ABDACDA
C)ADBACA
D)No Euler circuit exists.

Accepted Answer

The answer of Find an Euler circuit for the figure...

Quiz 7: Graph Theory

Suppose that a graph containing four vertices A, B, C, and D has an Euler circuit starting from A. If the degree of the vertex B is 4, then how many times will the Euler circuit pass the vertex B?

A circuit or cycle in a graph is a path that begins and ends at the same vertex.

What is the degree of each vertex in the figure below?

When graphs are represented pictorially using dots and segments, the dots are called ___________.

How many Euler circuits starting from the vertex B are possible?

Find an Euler circuit for the figure below.

According to Euler's theorem, the figure below has an Euler circuit.

The _______________ of a vertex is the number of edges that touch that vertex.

If a circuit traverses each edge of the graph exactly once, it is called a(n) _____________ circuit.

Suppose the edges of a certain graph represent phone lines that must be maintained, and the vertices represent junctions. A worker maintaining the lines would like to find an Euler circuit for this graph to increase ___________.

The computers in an office area are connected as follows:

What is the degree of vertex F?

Find an Euler circuit for the figure below.

Find an Euler circuit for the figure below.

What is the degree of each vertex in the figure below?

Which of the following vertices have degree 2?

According to Euler's theorem, a connected graph has an Euler circuit precisely when every vertex has a(n) ___________ degree.

Which of the following vertices have degree 2?

According to Euler's theorem, the figure below has an Euler circuit.

Find an Euler circuit for the figure below.