Pósa-Seymour Conjecture

Statement

In 1962, Lajos Pósa conjectured that every $n$-vertex graph with minimum degree at least $\frac{2n}3$ contains the square of a Hamiltonian cycle, generalizing Dirac's theorem. The conjecture was further extended by Paul Seymour in 1974 to the following:

Every graph on $n$ vertices with minimum degree at least $\frac{kn}{k+1}$ contains the $k$-th power of a Hamiltonian cycle.

The blow-up lemma was used by Komlós, Sárközy, and Szemerédi to prove the conjecture for all sufficiently large values of $n$ (for a fixed $k$) in 1998.