The Bandwidth Theorem in sparse graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $big(tfrac{k-1}{k}+o(1)big)n$ 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 $pgg big(tfrac{log n}{n}big)^{1/Delta}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $big(tfrac{k-1}{k}+o(1)big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Delta$, bandwidth $o(n)$, and at least $C p^{-2}$ 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 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 $pgg big(tfrac{log n}{n}big)^{1/3}$.