Almost spanning subgraphs of random graphs after adversarial edge removal

Let Δ⩾2 be a fixed integer. We show that the random graph Gn,p with p⩾c(logn/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 Gn,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.