Almost Spanning Subgraphs of Random Graphs After Adversarial Edge Removal

Abstract

Let Δ ≥ 2 be a fixed integer. We show that the random graph${\mathcal{G}_{n,p}}$with$p\gg (\log n/n)^{1/\Delta}$is robust with respect to the containment of almost spanning bipartite graphsHwith maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from${\mathcal{G}_{n,p}}$in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all suchH.