# Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs

@article{Noever2017LocalRF, title={Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs}, author={Andreas Noever and Angelika Steger}, journal={Electr. J. Comb.}, year={2017}, volume={24}, pages={P4.8} }

- Published in Electr. J. Comb. 2017

In 1962, Pósa conjectured that a graph G = (V,E) contains a square of a Hamiltonian cycle if δ(G) ≥ 2n/3. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi 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 ǫ > 0 and p = n−1/2+ǫ a.a.s. every subgraph of Gn,p with minimum degree at least (2/3 + ǫ)np contains the square of a cycle on (1 − o(1))n vertices. This is almost best possible in three ways… CONTINUE READING