2.1.1. Definition of Hamiltonian Circuit and Hamiltonian Path

Try these first

1. 1.

   Define Hamiltonian Circuit.

   Hint

   Think about a loop that goes around a graph.

2. 2.

   What is a Hamiltonian Path?

   Hint

   Consider it like a tour that doesn't have to loop back.

3. 3.

   What constitutes a Hamiltonian circuit?

   • Visits all edges
   • Visits all vertices exactly once
   • Can repeat vertices

   Hint

   Focus on the vertex requirement.

4. 4.

   True or False: An Euler circuit can also be a Hamiltonian circuit.

   • True
   • False

   Hint

   Think about what each circuit focuses on.

5. 5.

   Given a graph with a vertex degree distribution, prove whether it satisfies Dirac's or Ore's theorem or neither.

   Hint

   Compare degree sums and adjacency pairs carefully.

6. 6.

   Create your own small graph and identify Hamiltonian circuits or paths within it.

   Hint

   Use graphical tools or drawings for clarity.