Allen, PeterPeterAllenBöttcher, JuliaJuliaBöttcherEhrenmüller, JuliaJuliaEhrenmüllerTaraz, AnuschAnuschTaraz2020-11-092020-11-092020-05-15Advances in Combinatorics 1 (2020): 6, 1-60 (2020)http://hdl.handle.net/11420/7750The bandwidth theorem [Mathematische Annalen, 343(1):175–205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p ≫[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp−2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫[Formula Presented].en2517-559920201160Alliance of Diamond Open Access Journalshttps://creativecommons.org/licenses/by/3.0/MathematikJournal Article10.15480/882.304710.19086/aic.1284910.15480/882.30471612.00661v2Journal Article