An Eulerian trail is a trail with exactly 2 odd vertices and starts and ends on either of the odd vertices.
Check each network if they fit the categories for a Eulerian trail.
Start by counting the degrees of each network’s vertices.
1st Figure
This network does not have any odd vertices. Therefore, it does not have an Eulerian trail.
2nd Figure
This network has more than two odd vertices. Therefore, it does not have an Eulerian trail.
3rd Figure
This network does not have any odd vertices. Therefore, it does not have an Eulerian trail.
4th Figure
This network has exactly two odd vertices. Therefore, it may have an Eulerian trail.
Next, further check if the 4th figure has a Eulerian trail by drawing a diagram to check if the trail starts at one odd vertex and ends on the other odd vertex.
You can mark the starting vertex with S and the finishing vertex with F.
Either of the two diagrams illustrate that you can start on either odd vertex and end on the other. Therefore, the 4th figure fits both categories for a Eulerian trail.
Question 2 of 4
2. Question
Which of the following networks has an Eulerian trail?
An Eulerian trail is a trail with exactly 2 odd vertices and starts and ends on either of the odd vertices.
Check each network if they fit the categories for a Eulerian trail.
Start by counting the degrees of each network’s vertices.
1st Figure
This network has more than two odd vertices. Therefore, it does not have an Eulerian trail.
2nd Figure
This network has more than two odd vertices. Therefore, it does not have an Eulerian trail.
3rd Figure
This network has more than two odd vertices. Therefore, it does not have an Eulerian trail.
4th Figure
This network has exactly two odd vertices. Therefore, it may have an Eulerian trail.
Next, further check if the 4th figure has a Eulerian trail by drawing a diagram to check if the trail starts at one odd vertex and ends on the other odd vertex.
You can mark the starting vertex with S and the finishing vertex with F.
The diagram illustrates that you can start on one odd vertex and end on the other. Therefore, the 4th figure fits both categories for a Eulerian trail.
Question 3 of 4
3. Question
Which of the following networks has an Eulerian circuit?
An Eulerian circuit is a circuit with all of its vertices having even degrees, starts and ends on the same vertex, and passes on all the edges only once.
Check each network if they fit the categories for an Eulerian circuit.
Start by counting the degrees of each network’s vertices.
1st Figure
Not all vertices are even for this network. Therefore, it does not have an Eulerian circuit.
2nd Figure
Not all vertices are even for this network. Therefore, it does not have an Eulerian circuit.
3rd Figure
Not all vertices are even for this network. Therefore, it does not have an Eulerian circuit.
4th Figure
This network’s vertices all has even degrees. Therefore, it fits one category for an Eulerian circuit.
Next, further check if the 4th figure has an Eulerian circuit by drawing a diagram to check if the trail starts and ends on the same vertex and if it passes all the edges once.
You can mark the starting and finishing vertex with both S and F.
This diagram illustrates that you can start and finish on the same vertex and pass through all edges just once. Therefore, the 4th figure fits all categories for an Eulerian circuit.
Question 4 of 4
4. Question
The roads for a new mailbox delivery route is shown in this network diagram. The delivery will start and end at P and will travel along each road once. Answer the following questions:
Answer Y for yes or N for no
(i) Does an Eulerian circuit exist on the given network?=(Y, y)
An Eulerian circuit is a circuit with all of its vertices having even degrees, starts and ends on the same vertex, and passes on all the edges only once.
(i) Does an Eulerian circuit exist on the given network?
Check the network if they fit the categories for an Eulerian circuit.
Start by counting the degrees of each network’s vertices.
This network’s vertices all has even degrees. Therefore, it fits one category for an Eulerian circuit.
Next, further check if the figure has an Eulerian circuit by drawing a diagram to check if the trail starts and ends vertex P and if it passes all the edges once.
This diagram illustrates that you can start and finish on vertex P and pass through all edges just once. Therefore, an Eulerian trail exists for this network.
(ii) Can the Eulerian circuit start at U?
Check if the Eulerian circuit can start and end vertex U and if it passes all the edges once.
This diagram illustrates that a Eulerian circuit can start and finish on vertex U and pass through all edges just once.
(iii) Can the Eulerian circuit start at R?
Check if the Eulerian circuit can start and end vertex R and if it passes all the edges once.
This diagram illustrates that a Eulerian circuit can start and finish on vertex R and pass through all edges just once.
(i) Does an Eulerian circuit exist on the given network? Yes
