Eulerian and Hamiltonian paths/circuits
Eulerian
An Eulerian graph is a graph in which you can travel through every edge (line) exactly once and return to the starting point. In other words, the graph contains an Eulerian circuit.
Hamiltonian
A Hamiltonian graph is a graph that contains a Hamiltonian circuit—a path that visits every vertex (point) exactly once and returns to the starting vertex.
Eulerian and Hamiltonian
The main difference between Eulerian and Hamiltonian paths/circuits is what they must pass through.
Paths and Circuits
The main difference between a path and a circuit is where you end
1. Eulerian Path
An Eulerian Path is a path that uses every edge (line) exactly once.
You may visit the same vertex (point) more than once.
The important thing is do not repeat any edge.
Possible Eulerian Path:
A → B → C → A
2. Eulerian Circuit
An Eulerian Circuit is an Eulerian Path that starts and ends at the same vertex.
Possible Eulerian Circuit :
A → B → C → A
3. Hamiltonian Path
A Hamiltonian Path visits every vertex exactly once.
You may not visit the same vertex twice.
Not every edge has to be used.
Hamiltonian Path:
A → B → C → D
4. Hamiltonian Circuit
A Hamiltonian Circuit visits every vertex exactly once and returns to the starting vertex.
Hamiltonian Circuit:
A → B → C → D → A
Real-life Examples
A mail carrier delivers letters by walking through every street exactly once and finishes at a different post office. Answer: Eulerian Path
A maintenance worker inspects every hallway exactly once and returns to the main entrance. Answer: Eulerian Circuit
A tourist wants to visit every tourist attraction exactly once and finishes at the last attraction. Answer: Hamiltonian Path
A salesperson visits every city exactly once and returns to the city where the trip started. Answer: Hamiltonian Circuit