2.1.3. Ore's Condition

Try these first

1. 1.

   Define a Hamiltonian circuit.

   Hint

   Think about the criteria for visiting vertices in a closed loop.

2. 2.

   What is Ore's condition in relation to Hamiltonian graphs?

   Hint

   Consider the role of vertex degrees in forming connections.

3. 3.

   What is a Hamiltonian circuit?

   • A circuit that visits each vertex once
   • A circuit that visits some vertices
   • A circuit that can revisit vertices

   Hint

   A Hamiltonian tour avoids repetition.

4. 4.

   True or False: Ore's condition must always guarantee a Hamiltonian cycle.

   • True
   • False

   Hint

   Consider graphs that satisfy Ore's condition without forming cycles.

5. 5.

   Analyze a graph with vertices A, B, C, D, E arranged such that A-B, A-C, C-D, D-E are edges. Determine if Ore's condition holds and justify your conclusion.

   Hint

   Be methodical in your degree counts and remember the basics of degree sums.

6. 6.

   Design a minimal graph of 7 characters where Ore's condition is satisfied, but the graph is not Hamiltonian. Explain your configuration.

   Hint

   Focus on distributing vertex connections strategically.