We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(varepsilon,p,k,ell)$-pseudorandom if for all disjoint $X$ and $Ysubset V(G)$ with $|X|gevare
psilon p^kn$ and $|Y|gevarepsilon p^ell n$ we have $e(X,Y)=(1pmvarepsilon)p|X||Y|$. We prove that for all $beta>0$ there is an $varepsilon>0$ such that an $(varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $beta pn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,lambda)$-graphs with $lambdall d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szabo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counti
Kuhn, Osthus and Taraz showed that for each gamma>0 there exists C such that any n-vertex graph with minimum degree gamma n contains a planar subgraph with at least 2n-C edges. We find the optimum value of C for all gamma<1/2 and sufficiently large n.
By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erd~os-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-
1)-partite graph H whose smallest part has t vertices, and any fixed c>0, there exists a constant C such that whenever G is an n-vertex graph with minimum degree at least ((3r-4)/(3r-1)+c)n, either G contains H, or we can delete at most Cn^(2-1/t) edges from G to yield an r-partite graph.
