On prisms, Möbius ladders and the cycle space of dense graphs
European Journal of Combinatorics
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In 1962 Pósa conjectured that every graph G on n vertices with minimum degree \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\delta(G)\ge \frac{2}{3}n\end{align*} \end{document} **image** contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3}n\end{align*} \end{document} **image** . Still in 1996, Komlós, Sárközy, and Szemerédi proved Pósa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n ≥ n0, where n0 is a very large constant. Here we show without using these lemmas that n0:= 2 × 108 is sufficient. We are motivated by the recent work of Levitt, Sárközy and Szemerédi, but our methods are based on techniques that were available in the 90's. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.