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?

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.