Pósa's theorem

Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small number of low-degree vertices. It is named after Lajos Pósa, a protégé of Paul Erdős born in 1947, who discovered this theorem in 1962.

The Pósa condition for a finite undirected graph $$G$$ having $$n$$ vertices requires that, if the degrees of the $$n$$ vertices in increasing order as
 * $$d_{1} \leq d_{2} \leq ... \leq d_{n},$$

then for each index $$k < n/2$$ the inequality $$k < d_{k}$$ is satisfied.

Pósa's theorem states that if a finite undirected graph satisfies the Pósa condition, then that graph has a Hamiltonian cycle in it.