No, that's the correct definition of a Hamiltonian cycle. However, the article isn't as clear as it could be about what the "increasingness" aspect applies to.

No, that's the correct definition of a Hamiltonian cycle. However, the article isn't as clear as it could be about what the "increasingness" aspect applies to. > It’s possible to think about any property, so long as it is “increasing” — that is, if adding more edges to a graph that already contains the property will not destroy the property. The "property" here isn't the Hamiltonian cycle itself, it's the property "a Hamiltonian cycle exists within this graph". This property, rather than the cycle itself, is increasing: if you add edges to a graph that contains a Hamiltonian cycle it will still contain a Hamiltonian cycle.