Practice Dirac's Theorem (2.1.2) - Hamiltonian Circuit - Discrete Mathematics - Vol 3

Practice: Dirac's Theorem

Lesson 2.1.2Hamiltonian Circuit

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9 questions

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6 sample questions

Question 1.
What is a Hamiltonian circuit?
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Think about paths that start and end at the same point.
Question 2.
State Dirac's Theorem.
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Focus on vertex degrees in relation to the number of vertices.
3Question 3.
What does Dirac's Theorem state?
- The sum of the degrees of non-adjacent vertices is always less than n.
- A connected graph is Hamiltonian if every vertex has degree at least n/2.
- All Eulerian graphs are Hamiltonian.
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Focus on the degree condition related to connected graphs.
4Question 4.
True or False: Ore's condition is more restrictive than Dirac's.
- True
- False
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Consider the flexibility in vertex degree comparisons.
Question 5.
Given a connected graph of 8 vertices where each vertex has a degree of 4, is the graph Hamiltonian? Explain with reference to Dirac's Theorem.
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Check if the vertex degrees meet the necessary condition.
Question 6.
Construct a graph with 6 vertices that satisfies Ore's condition yet lacks a Hamiltonian circuit. Demonstrate how.
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Consider what non-adjacency and degree sums indicate toward connectivity.

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