Definition
A Hamiltonian path is a path that visits all vertices of a graph exactly once.
A Hamiltonian cycle is a Hamiltonian path that, in addition, returns to the starting vertex, forming a cycle.
Summary:
Hamiltonian path = visits all vertices once.
Hamiltonian cycle = Hamiltonian path + returns to the start.
Hamiltonian path = visits all vertices once.
Hamiltonian cycle = Hamiltonian path + returns to the start.
Examples
Below are two typical examples: a graph that has a Hamiltonian cycle and one that does not have.
Example A — Has a Hamiltonian cycle
One possible cycle: A → B → C → D → A.
Example B — Does not have a Hamiltonian cycle
Vertex F has degree 1, so a Hamiltonian cycle cannot exist.