On a sparse random graph with minimum degree three: Likely Pósa sets are large

We consider the endpoint sets produced by Pósa rotations, when applied to a longest path in a random graph with cn edges, conditioned on having minimum degree at least three. We prove that, for c≥2.7, the Pósa sets are likely to be almost linear in n, implying that the number of missing edges, each allowing either to get a longer path or to form a Hamilton cycle, is almost quadratic in n.