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}
}
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

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