Abstract
We study the model (Formula presented.) of randomly perturbed dense graphs, where (Formula presented.) is any n-vertex graph with minimum degree at least (Formula presented.) and (Formula presented.) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every (Formula presented.) and (Formula presented.), and every n-vertex graph F with maximum degree at most Δ, we show that if (Formula presented.), then (Formula presented.) with high probability contains a copy of F. The bound used for p here is lower by a log -factor in comparison to the conjectured threshold for the general appearance of such subgraphs in (Formula presented.) alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the first example of graphs where the appearance threshold in (Formula presented.) is lower than the appearance threshold in (Formula presented.) by substantially more than a log -factor. We prove that, for every (Formula presented.) and (Formula presented.), there is some (Formula presented.) for which the kth power of a Hamilton cycle with high probability appears in (Formula presented.) when (Formula presented.). The appearance threshold of the kth power of a Hamilton cycle in (Formula presented.) alone is known to be (Formula presented.), up to a log -term when (Formula presented.), and exactly for (Formula presented.). © 2020 The Authors. Mathematika published by John Wiley & Sons Ltd on behalf of University College London.