## Proof of the Pósa−Seymour Conjecture

In 1974 Paul Seymour conjectured that any graph *G* of order *n* and minimum degree at least *(k−1)/k · n* contains the *(k − 1)*^{th} power of a Hamiltonian cycle. This conjecture was proved with the help of the Regularity Lemma – Blow-up Lemma method for *n ≥ n*_{0} where *n*_{0} is very large. Here we present another proof that avoids the use of the Regularity Lemma and thus the resulting *n*_{0} is much smaller. The main ingredient is a new kind of connecting lemma.

Joint work with Asif Jamshed.