P\'osa's Conjecture for graphs of order at least 2\times 10^8
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.
Keywords
Cite
@article{arxiv.1104.4367,
title = {P\'osa's Conjecture for graphs of order at least 2\times 10^8},
author = {Phong Châu and Louis DeBiasio and H. A. Kierstead},
journal= {arXiv preprint arXiv:1104.4367},
year = {2011}
}
Comments
20 pages, to appear in Random Structures & Algorithms