To analyze this question, let's break it into two parts:

Definition of a Hamiltonian Circuit

A Hamiltonian circuit is a cycle in a graph that visits each vertex exactly once and returns to the starting vertex. It must:

• Cover all vertices exactly once.
• Start and end at the same vertex.

Does this graph have a Hamiltonian Circuit?

The graph provided has 6 vertices labeled $1, 2, 3, 4, 5, 6$. To check if it has a Hamiltonian circuit:

1. Check the structure: The graph is connected, meaning every vertex is reachable from any other vertex.
2. Try to find a Hamiltonian circuit by inspection:
   • Starting at vertex $1$, we can follow this path: $1 \to 2 \to 4 \to 6 \to 5 \to 3 \to 1$
   • This path visits all vertices exactly once and returns to the starting vertex $1$. Hence, this is a Hamiltonian circuit.

Conclusion

Yes, the graph contains a Hamiltonian circuit. One example of such a circuit is: $1 \to 2 \to 4 \to 6 \to 5 \to 3 \to 1$

Would you like more details or clarification? Let me know!

Follow-up questions:

1. What are the conditions for a graph to have a Hamiltonian circuit?
2. How does a Hamiltonian circuit differ from an Eulerian circuit?
3. Can a graph have more than one Hamiltonian circuit? If so, how can you find them all?
4. Are there algorithms to determine if a Hamiltonian circuit exists? What are they?
5. How does the degree of vertices in a graph affect the existence of Hamiltonian circuits?

Tip:

To verify if a graph contains a Hamiltonian circuit, try tracing paths manually or use algorithms like the backtracking approach for systematic checks.