P\'osa's Conjecture for graphs of order at least 2\times 10^8

TL;DR

This paper proves Pósa's Conjecture for graphs with at least 200 million vertices using methods from the 1990s, avoiding the complex Regularity and Blow-up Lemmas.

Contribution

It establishes the conjecture for large graphs without relying on advanced lemmas, reducing the threshold to 2×10^8 vertices.

Findings

Pósa's Conjecture holds for graphs with ≥ 2×10^8 vertices.

Avoids using Regularity and Blow-up Lemmas in the proof.

Provides a more accessible proof for large graphs.

Abstract

In 1962 P\'osa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of P\'osa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 2n/3. Still in 1996, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved P\'osa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n > n_0, where n_0 is a very large constant. Here we show without using these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the recent work of Levitt, Szemer\'edi and S\'ark\"ozy, but our methods are based on techniques that were available in the 90's.