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To determine which graphs have Hamiltonian circuits, we need to check if each graph contains a Hamiltonian circuit. A **Hamiltonian circuit** is a path in a graph that visits each vertex exactly once and returns to the starting vertex.

Let's analyze the graphs:

- **Graph A:** This graph has several vertices and edges, but upon inspection, it doesn't appear to have a path that visits every vertex exactly once without repeating and returns to the starting point. Hence, no Hamiltonian circuit.

- **Graph B:** This graph looks quite connected and upon closer inspection, it does have a Hamiltonian circuit. You can traverse the graph in a way that visits each vertex exactly once and returns to the starting point.

- **Graph C:** This graph also does not seem to have a Hamiltonian circuit because there doesn't appear to be a way to visit every vertex exactly once without repeating edges or vertices.

- **Graph D:** This graph is sparse in connections. It's highly unlikely for a Hamiltonian circuit to exist since many vertices are not connected in a way that allows a complete traversal.

**Conclusion**:
- **Graph B** has a Hamiltonian circuit.

Would you like further details or explanations?

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Here are five related questions for deeper exploration:
1. What is the difference between a Hamiltonian circuit and an Eulerian circuit?
2. Can a graph with disconnected components have a Hamiltonian circuit?
3. What are the necessary conditions for a graph to contain a Hamiltonian circuit?
4. How does the degree of vertices influence the existence of Hamiltonian circuits?
5. Can a Hamiltonian circuit exist in a directed graph? How would you check?

**Tip**: When checking for Hamiltonian circuits, try visualizing a continuous path that visits all vertices exactly once and returns to the start, while avoiding repeating vertices.

Which of the following graphs have Hamiltonian circuits?

Analyze a set of graphs to determine which ones contain Hamiltonian circuits, a key concept in graph theory. Suitable for high school students learning advanced graph theory concepts.

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