Local resilience for squares of almost spanning cycles in sparse random graphs

In 1962, P\'osa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices

Stylised facts of economic growth in developing countries

This paper offers a concise survey on the literature of growth empirics applying to DCs. It is argued that there is a number of important stylised facts of economic growth relevant to DCs which are not included in the corresponding lists of Kaldor and Romer. In contrary to the usual procedure, the growth rates of per capita income are calculated by employing potential output, which is determined by the use of the Hodrick-Prescott-filter. Finally, three important conclusions resulting from the empirical observations are discussed in the last section. --Stylised Facts,Economic Growth,Developing Countries,Growth Empirics