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arXiv · 1104.4367

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

Phong Châu·Louis DeBiasio·H. A. Kierstead
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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.

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Phong Châu, Louis DeBiasio, H. A. Kierstead. 2011-04-21. P\'osa's Conjecture for graphs of order at least 2\times 10^8. https://arxiv.org/abs/1104.4367

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