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 C 0 such that for all p satisfying p ( n ) ≥ C 0 / 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. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011