Proof of the Pósa–Seymour Conjecture

In 1974 Paul Seymour conjectured that any graph $G$ of order $n$ and minimum degree at least $(k-1)/k \cdot n$ contains the $(k-1)^{th}$ power of $k$ a Hamiltonian cycle. This conjecture was proved with the help of the Regularity Lemma –– Blow-up Lemma method for $n \geq n_0$ where $n_0$ is very large. Here we present another proof that avoids the use of the Regularity Lemma and thus the resulting n0 is much smaller. The main ingredient is a new kind of connecting lemma.
Joint work with Asif Jamshed