In this paper we investigate how the use of the Regularity Lemma and the Blow- up Lemma can be avoided in certain extremal problems of dense graphs. We present the ideas for the following well-known Pósa conjecture on the square of a Hamiltonian cycle. In 1962 Pósa conjectured that any graph G of order n and minimum degree at least 2/3n contains the square of a Hamiltonian cycle. In an earlier paper we proved this conjecture with the use of the Regularity Lemma-Blow-up Lemma method for n ¸ n0 where n0 is very large. Here we present another proof (and a general method) that avoids the use of the Regularity Lemma and thus the resulting n0 is much smaller.
, Sárközy, Gábor N.
, Szemerédi, Endre
(2008). How to avoid using the Regularity Lemma; Pósa's Conjecture revisited. .
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