Question 1

How many edges does the graph have?

Accepted Answer

By counting the number of lines connecting the vertices, we can see that there are 6 edges in the graph. The graph has 6 edges. It can be counted by simply looking at the lines connecting the vertices.

Question 2

Find a Hamiltonian path.

Accepted Answer

A

Question 3

List the vertices of the graph.

Accepted Answer

The graph includes all six vertices, A, B, C, D, E, and F. Therefore, option D is the best choice. The graph has six vertices labeled A, B, C, D, E, and F, so the correct answer is D.

Question 4

Suppose you're trying to make a final exam schedule for six classes: anthropology 107 (A),
Economics 212 (E), information systems 194 (I), statistics 261 (S), history 201 (H), and geology 101
(G). The graph below shows which classes students have in common: these are connected by edges.
Use graph coloring to find the chromatic number.

Accepted Answer

Using graph coloring, we need to assign colors to each node (class) such that no two adjacent nodes have the same color. Starting with the node with the highest degree (most connections), we can assign a color and work our way through the graph until all nodes have a color. In this case, the node with the highest degree is G (geology 101) with connections to A, S, and H.

We can assign G the color 1 and then move on to its neighbors. A and S are connected, so they must have different colors. We can assign A the color 2 and S the color 3. Now, we move on to H, which is only connected to G, so it can have the same color as G (color 1).

Moving on to the next highest degree node, we have E (Economics 212), which is connected to I and S. We already used color 3 for S, so we need to use a different color for E. We can use color 2 for E and color 4 for I.

Finally, we have one node left, which is H (history 201). H is only connected to G (color 1), so it can have a different color, such as color 5.

Thus, we can color the graph with 4 colors, so the chromatic number is 4. Therefore, the answer is C. The chromatic number is the minimum number of colors needed to color the graph such that no two adjacent vertices have the same color. Using graph coloring, we can see that we need at least 4 colors to properly color this graph. Therefore, the answer is C.

Question 5

Find the number of Hamilton circuits if a complete graph has seven vertices.

Accepted Answer

The answer of Find the number of Hamilton circuits if...

Question 6

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an
Euler path, or neither an Euler circuit nor an Euler path.
The graph has 4 odd vertices and 2 even vertices.

Accepted Answer

A graph with an Euler circuit must have all even vertices, while a graph with an Euler path must have exactly 2 odd vertices (or 0 odd vertices if it is a connected graph). Since this graph has 4 odd vertices, it cannot have an Euler path, and since it has 2 even vertices, it cannot have an Euler circuit. Therefore, the answer is neither an Euler circuit nor an Euler path. A connected graph with an even number of vertices has an Euler circuit if and only if each vertex has even degree. In this case, since there are 4 odd vertices, there is no Euler circuit. Similarly, a connected graph has an Euler path if and only if it has zero or two vertices with odd degree. Since there are 4 odd vertices in this graph, there is no Euler path either.

Question 7

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an
Euler circuit, find one.

Accepted Answer

The graph has odd degree vertices (H, G, F) which means it cannot have an Euler circuit. However, it does have an Euler path that starts and ends at vertex H, which is provided in choice B. Choices C and D incorrectly state that the graph has an Euler circuit. Choice A correctly states that the graph has neither an Euler path nor an Euler circuit. The graph has an Euler path, as shown in option B. It starts and ends at vertex H and visits every edge exactly once. The other options either incorrectly state that there is an Euler circuit when there is not (option C) or omit part of the path (option D) or falsely claim that there is neither an Euler path nor an Euler circuit (option A).

Question 8

Represent the figure using a graph. Use vertices for islands and edges for bridges.

Accepted Answer

Choice B represents the figure accurately using vertices to represent the islands and edges to represent the bridges. Choices A, C, and D do not accurately represent the figure.

Question 9

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area
And edges to represent the connecting doors. Determine if the graph has an Euler path, and Euler
Circuit, or neither. If the graph has an Euler path or an Euler circuit, find one.

Accepted Answer

The answer of Draw a graph that represents the floor...

Question 10

Find a circuit that includes vertex D.

Accepted Answer

The circuit must start and end at vertex D, so options A and D are not valid. Option B does not form a closed circuit because it does not return to vertex D. Option C includes all the vertices (A,B,C,D) and forms a closed circuit starting and ending at D. Therefore, it is the best choice.

Question 11

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an
Euler path, or neither an Euler circuit nor an Euler path.
The graph has 2 odd vertices and 3 even vertices.

Accepted Answer

A graph has an Euler path if and only if it has exactly 2 odd vertices.

Question 12

Determine if an Euler path or an Euler circuit exists so that a person who plows the roads does not
Have to pass over any street twice. If an Euler path or an Euler circuit exists, find one. The
Intersections of the streets have been labeled for you.

Accepted Answer

The answer of Determine if an Euler path or an...

Question 13

Using graph coloring, find the smallest number of colors needed to color the map so that no regions
Sharing a common border are the same color

Accepted Answer

The answer of Using graph coloring, find the smallest number...

Question 14

Are the two graphs equivalent?

Accepted Answer

The answer of Are the two graphs equivalent? 
...

Question 15

List the edges that are bridges.

Accepted Answer

The answer of List the edges that are bridges. 
...

Question 16

Find a Hamiltonian circuit.

Accepted Answer

The answer of Find a Hamiltonian circuit. 
...

Question 17

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an
Euler circuit, find one.

Accepted Answer

The graph has vertices of odd degree (A, C, E, G) which means it cannot have an Euler path or an Euler circuit.

Question 18

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area and edges to
Represent the connecting doors.

Accepted Answer

The answer of Draw a graph that represents the floor...

Question 19

List the even vertices of the graph.

Accepted Answer

Even vertices are the ones with an even degree, which means they are connected to an even number of edges. In this graph, only B, C, D, and E have an even degree, as each one of them is connected to two edges. Therefore, the correct choice is D. Even vertices are those which have an even degree. In the given graph, vertices B, C, D, and E have an even degree of 2. Therefore, the answer is D. A and C are incorrect as they have an odd degree of 3 and 1 respectively. B is incorrect as it includes all the vertices of the graph, not just the even ones.

Question 20

Draw a graph for the figure using vertices for the islands and edges for the bridges. Determine if the
Graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or Euler circuit, find one.

Accepted Answer

The answer of Draw a graph for the figure using...

Quiz 13: Graph Theory

How many edges does the graph have?

Find a Hamiltonian path.

List the vertices of the graph.

Find the number of Hamilton circuits if a complete graph has seven vertices.

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an Euler path, or neither an Euler circuit nor an Euler path. The graph has 4 odd vertices and 2 even vertices.

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an Euler circuit, find one.

Represent the figure using a graph. Use vertices for islands and edges for bridges.

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area And edges to represent the connecting doors. Determine if the graph has an Euler path, and Euler Circuit, or neither. If the graph has an Euler path or an Euler circuit, find one.

Find a circuit that includes vertex D.

Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an Euler path, or neither an Euler circuit nor an Euler path. The graph has 2 odd vertices and 3 even vertices.

Determine if an Euler path or an Euler circuit exists so that a person who plows the roads does not Have to pass over any street twice. If an Euler path or an Euler circuit exists, find one. The Intersections of the streets have been labeled for you.

Using graph coloring, find the smallest number of colors needed to color the map so that no regions Sharing a common border are the same color

Are the two graphs equivalent?

List the edges that are bridges.

Find a Hamiltonian circuit.

State whether the graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or an Euler circuit, find one.

Draw a graph that represents the floor plan. Use vertices to represent the rooms and outside area and edges to Represent the connecting doors.

List the even vertices of the graph.

Draw a graph for the figure using vertices for the islands and edges for the bridges. Determine if the Graph has an Euler path, an Euler circuit, or neither. If it has an Euler path or Euler circuit, find one.