2.1.4. Proof of Ore's Theorem

Try these first

1. 1.

   Define a Hamiltonian circuit.

   Hint

   Think of a tour that starts and ends at the same vertex.

2. 2.

   What does Dirac's theorem specify regarding vertex degrees?

   Hint

   Recall the minimum degree condition.

3. 3.

   What is a Hamiltonian circuit?

   • A path visiting all edges
   • A path visiting all vertices
   • A path visiting some vertices

   Hint

   Think about what must be visited in a circuit.

4. 4.

   True or False: Dirac's Theorem is both necessary and sufficient for a graph to be Hamiltonian.

   • True
   • False

   Hint

   Consider examples that defy this condition.

5. 5.

   Design a graph with 6 vertices that adheres to Ore's condition and prove it is Hamiltonian.

   Hint

   Start with vertices connected linearly, then add links that do not disrupt the required sums.

6. 6.

   Prove or disprove Ore's theorem on a bipartite graph by illustrating vertex pairs.

   Hint

   Consider how bipartite graphs are structured and whether all vertex combinations meet Ore's condition.