A graph is Hamiltonian if it has a Hamiltonian cycle.

A useful condition both necessary and sufficient for a graph to be Hamiltonian is not known. Ore's theorem and the Bondy and Chvátal theorem give sufficient conditions, while a necessary condition follows quickly from the definition, namely:

Let be a graph of order at least 3. If is Hamiltonian, then for every proper subset of , the subgraph induced by has at most components.