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arXiv · 1203.6310

On Posa's conjecture for random graphs

Daniela Kühn·Deryk Osthus
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Abstract

The famous Posa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Komlós, Sarközy and Szemerédi. Here we prove that if p > n^{-1/2+\eps}, then asymptotically almost surely, the binomial random graph G_{n,p} contains the square of a Hamilton cycle. This provides an `approximate threshold' for the property in the sense that the result fails to hold if p< n^{-1/2}.

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Daniela Kühn, Deryk Osthus. 2012-07-30. On Posa's conjecture for random graphs. https://arxiv.org/abs/1203.6310

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