Almost spanning subgraphs of random graphs after adversarial edge removal

TL;DR

This paper proves that random graphs with certain edge probabilities are resilient to adversarial edge removals, still containing almost spanning bipartite subgraphs with bounded degree and sublinear bandwidth.

Contribution

It establishes robustness of G(n,p) against adversarial edge deletions for embedding almost spanning bipartite graphs with bounded degree and sublinear bandwidth.

Findings

G(n,p) remains containing all such bipartite graphs after adversarial deletions

Edge probability threshold p>>(log n/n)^{1/Delta} is critical for robustness

Resilience holds even when each vertex loses less than half of its neighbors

Abstract

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