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) Can Jacob deliver the pizzas while passing all the streets once?
The previous question confirmed that we can use the sequence A−F−D−C−B−A−E−F−B−D−E to pass through all the streets once.
Therefore, Jacob can deliver the pizza while passing all the streets only once.
(ii) Can Jacob start and finish on the same place?
Check if the sequence is an Eulerian trail or circuit by counting the edges of each vertex.
This network has exactly 2 odd edges and start at one of the odd edges and ends on the other. Therefore, it is an Eulerian trail.
Since an Eulerian trail starts and finishes on two different edges, this means that Jacob cannot start and finish at the same place.
(i) Can Jacob deliver the pizzas while passing all the streets once? Yes
(ii) Can Jacob start and finish on the same place? No
Question 3 of 4
3. Question
Moe delivers leaflets to each location marked on the network below. Can he deliver leaflets to all the locations while passing on each street once?
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.
For Moe to deliver the leaflets to all locations by passing all the streets only once, the network should have either an Eulerian trail or Eulerian circuit.
Check if the sequence is an Eulerian trail or circuit by counting the edges of each vertex.
This network has exactly 2 odd edges and start at one of the odd edges and ends on the other. Therefore, it is an Eulerian trail.
Next, further check if a Eulerian trail can be used to in this network.
You can mark the starting vertex with S and the finishing vertex with F.
The diagram illustrates that you can start on vertex T and end on vertex S. This means that the network has an Eulerian trail.
Therefore, Moe can deliver leaflets in all locations while passing all the streets once.
Question 4 of 4
4. Question
Moe delivers leaflets to each location marked on the network below. He discovered a new path from T to S. Which sequence can he take so he can start and finish from the same place?
