Almost spanning subgraphs of random graphs after adversarial edge removal

Let Δ ⩾ 2 be a fixed integer. We show that the random graph G n , p with p ⩾ c ( log n / n ) 1 / Δ is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Δ and sublinear bandwidth in the following sense. If an adversary deletes arbitrary edges in G n , p such that each vertex loses less than half of its neighbours, then asymptotically almost surely the resulting graph still contains a copy of H.