What is the difference between an Eulerian path and a Hamiltonian path?

What is the difference between an Eulerian path and a Hamiltonian path?

An Eulerian path visits every edge of a graph exactly once, while a Hamiltonian path visits every vertex exactly once.

Can you give me an example of each?

Sure! In a square, you can have an Eulerian path by tracing the edges without lifting your pen. A Hamiltonian path would be visiting each corner of the square without revisiting any.

Are there specific conditions for a graph to have these paths?

Yes! A graph has an Eulerian path if it has at most two vertices of odd degree. A Hamiltonian path doesn't have a simple condition, but it often depends on the graph's structure.

How do these concepts apply in real-world scenarios?

They are used in routing problems, like the traveling salesman problem for Hamiltonian paths and network design for Eulerian paths.