Hamiltonian Paths
Visit every vertex exactly once.
Key idea
Visit every vertex exactly once.
In depth
A Hamiltonian path visits every vertex exactly once. Unlike Euler paths there is no simple criterion; deciding existence is NP-complete. Sufficient conditions exist, such as Dirac’s: each vertex has degree ≥ n/2.
Worked example
Every complete graph Kₙ (n ≥ 3) has a Hamiltonian cycle.
Real-world uses
- Planning a tour that visits each city exactly once, such as a delivery route
Common mistake
Assuming a simple degree test exists as for Euler paths.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Hamiltonian Paths?
Visit every vertex exactly once. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Hamiltonian Paths used in real life?
Real-world examples: Planning a tour that visits each city exactly once, such as a delivery route.
What should I learn before Hamiltonian Paths?
Make sure you are comfortable with Euler Paths and Circuits, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Hamiltonian Paths?
Assuming a simple degree test exists as for Euler paths.
How can I practise Hamiltonian Paths?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Hamiltonian Paths?
Move on to Trees next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.