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

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, Sárközy, 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 > 0 and p = n−1/2+ a.a.s. every subgraph of Gn,p with minimum degree at least (2/3+ )np contains the square of a cycle on (1−o(1))n vertices. This is almost best possible in three ways: (1) for p 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. Gn,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.