To determine if a Hamiltonian circuit exists for the graph in Exercise 6 from the image:

Step 1: Hamiltonian Circuit Definition

A Hamiltonian circuit is a closed path that visits each vertex exactly once and returns to the starting vertex.

Step 2: Graph Analysis for Exercise 6

The graph in Exercise 6 has:

• 12 vertices labeled $1, 2, \dots, 12$
• A rectangular shape with multiple edges connecting vertices, resembling a grid structure.

To check for a Hamiltonian circuit, I need to determine whether there exists a path visiting all vertices exactly once and returning to the starting vertex.

Step 3: Trial and Error Approach

By trial and error, here is one possible Hamiltonian circuit:

Starting from vertex $1$, trace the path:
$1 \to 2 \to 3 \to 4 \to 8 \to 12 \to 11 \to 10 \to 9 \to 5 \to 6 \to 7 \to 1$.

Path Explanation:

1. The path starts at vertex $1$.
2. It visits each vertex exactly once.
3. It returns to the starting vertex $1$, forming a closed path.

Conclusion:

A Hamiltonian circuit exists for the graph in Exercise 6.

Would you like a diagram of the path or more details? Let me know!

Related Questions:

1. What is the difference between Hamiltonian and Eulerian paths?
2. Can you provide the adjacency matrix for Exercise 6?
3. How do we identify Euler paths in a graph?
4. What are the conditions for a graph to have a Hamiltonian circuit?
5. Are Hamiltonian circuits unique for a graph?

Tip: To identify Hamiltonian circuits, ensure you systematically check paths without repeating vertices. For larger graphs, algorithms like backtracking help.