Pósa's theorem

Let a finite simple graph on $n\ge 3$ vertices have nondecreasing degree sequence $d_1\le \cdots \le d_n$. Pósa's theorem gives the sufficient conditionfor a Hamiltonian cycle. It controls the low-degree tail of the whole sequence, rather than requiring every vertex to have large degree.

For example, any graph satisfying Dirac's theorem, with minimum degree at least $n/2$, satisfies this condition. The condition is sufficient rather than necessary: the cycle $C_6$ is Hamiltonian but has $d_2=2$. Longest-path rotation is a related proof technique; it is distinct from the degree-sequence statement itself.