2.2. Summary and References

Try these first

1. 1.

   What is a Hamiltonian circuit?

   Hint

   Think about circuits that don't repeat any vertex.

2. 2.

   What is the key difference between a Hamiltonian path and a Hamiltonian circuit?

   Hint

   Consider whether the end point is the same as the start.

3. 3.

   What is a key requirement for a Hamiltonian circuit?

   • It must cover all edges of the graph.
   • It can repeat vertices.
   • It must cover all vertices without repeats.

   Hint

   Think about circuits and their characteristics.

4. 4.

   Can a graph have a Hamiltonian circuit and not satisfy Dirac's Theorem?

   • True
   • False

   Hint

   Consider what 'sufficient' means in context.

5. 5.

   Given a counterexample of a graph that meets Ore’s condition but lacks a Hamiltonian circuit. Explain why it fails.

   Hint

   Reflect on vertex connectivity and arrangement.

6. 6.

   Design a graph that is Hamiltonian, shows Dirac’s theorem being applicable, but not Ore’s theorem. Describe the graphs.

   Hint

   Examine degree distributions and connectivity.