What are Hamiltonian and Eulerian paths?

Can you explain what Hamiltonian and Eulerian paths are?

Sure! A Hamiltonian path visits each vertex exactly once, while an Eulerian path visits each edge exactly once.

Can you give me an example of a Hamiltonian path?

Absolutely! In a triangle graph with vertices A, B, and C, a Hamiltonian path could be A -> B -> C.

What about an example of an Eulerian path?

In a square graph with vertices A, B, C, and D, an Eulerian path could be A -> B -> C -> D -> A.

How do I determine if a graph has a Hamiltonian or Eulerian path?

For Hamiltonian paths, there's no simple test, but for Eulerian paths, a graph has one if it has at most two vertices of odd degree.