Local resilience of almost spanning trees in random graphs

We prove that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n, p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1− α)n vertices, even after we delete a (1/2− γ)-fraction of the edges incident to each vertex. The proof uses Szemerédi’s regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs.