State the definition of an:
Euler trail
Euler circuit
For a connected network, state the condition that ensures that there is an:
Euler circuit
Euler trail but no Euler circuit
Suppose G is a very large network, with 300 vertices, each of even degree.
Determine if the following exists for G. Support your answer.
Euler trail
Euler circuit
For each of the following networks, determine whether it is possible to draw an Euler trail:
For each of the following networks, write an Euler trail starting with the indicated vertex:
Vertex A
Vertex B
For each of the following networks, write an Euler circuit starting with the indicated vertex:
Vertex A
Vertex A
Vertex B
Vertex D
Consider the given network:
Determine whether the network has an Euler trail and/or Euler circuit.
Using Fleury's algorithm, which vertex can the trail start at?
Noah wants to trace an Euler trail through the network that starts at the vertex D. According to Fleury's algorithm, which vertices are possible to travel to first along such a trail?
Write an Euler trail starting with the vertex D.
Consider the given network:
Determine whether the network has an Euler trail and/or Euler circuit.
Using Fleury's algorithm, which vertex can the trail start at?
Beth has started to trace an Euler trail through the graph, by moving along the vertices B, C, D. According to Fleury's algorithm, which vertices are possible to travel to next along such a trail?
Write an Euler trail for the network that completes Beth's trail.
Consider the given network:
Determine whether the network has an Euler trail and/or Euler circuit.
Using Fleury's algorithm, which vertex can an Euler trail start at?
Glen has started to trace an Euler circuit through the network, by moving along the vertices B, A, D, E, C, D, G. According to Fleury's algorithm, which vertices could Glen travel to next?
Write down the remaining vertices to complete the Euler trail for Glen:
B, A, D, E, C, D, G, \ldots
Determine whether the following networks are traversable:
Determine whether the following graphs are Euler networks:
Determine if the following networks are traversable but do not contain an Euler circuit:
For each of the following networks:
State the degree of each vertex.
State whether it is an Euler network.
For each of the following tables, state whether the corresponding network contains an Euler trail, an Euler circuit or neither:
| Vertex | Degree |
|---|---|
| A | 2 |
| B | 2 |
| C | 2 |
| D | 4 |
| E | 2 |
| F | 2 |
| Vertex | Degree |
|---|---|
| A | 2 |
| B | 3 |
| C | 2 |
| D | 3 |
| E | 2 |
| Vertex | Degree |
|---|---|
| A | 2 |
| B | 2 |
| C | 2 |
| D | 2 |
| Vertex | Degree |
|---|---|
| A | 2 |
| B | 3 |
| C | 5 |
| D | 3 |
| E | 2 |
A neighborhood watch group wants to plan their security route using the network below:
| Vertex | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Residence | \text{Jack} | \text{Justin} | \text{David} | \text{Paul} | \text{Yuri} | \text{Noah} | \text{Dylan} | \text{Carl} |
The route needs to be fuel efficient, by covering every street only once before moving to the next area.
Is this route possible?
The group wants to start at Justin's residence, then pick up Yuri, and finish at David's residence. Write an Euler trail for this.
A street cleaning crew was using the following map to plan the route they will follow:
They want to plan an efficient run by going over every street only once, starting and ending at the same point. Is this plan possible?
Determine whether the following circuits satisfy their conditions for an efficient route.
A, B, C, E, D, F, E, C, B, F, D, E, A
C, D, E, C, A, F, E, A, B, F, E, D, C
B, A, F, E, A, C, E, D, C, B, D, F, B
A, F, E, A, C, D, E, C, B, F, D, B, A