Embedding Spanning Bounded Degree Graphs In Randomly Perturbed Graphs

We study the model G alpha G(n,p) of randomly perturbed dense graphs, where G alpha is any n-vertex graph with minimum degree at least alpha n and G(n,p) 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 alpha >0 and Delta >= 5, and every n-vertex graph F with maximum degree at most Delta, we show that if p=omega (n-2/(Delta +1)), then G alpha G(n,p) 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 G(n,p) 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 G alpha G(n,p) is lower than the appearance threshold in G(n,p) by substantially more than a log-factor. We prove that, for every k >= 2 and alpha >0, there is some eta >0 for which the kth power of a Hamilton cycle with high probability appears in G alpha G(n,p) when p=omega (n-1/k-eta). The appearance threshold of the kth power of a Hamilton cycle in G(n,p) alone is known to be n-1/k, up to a log-term when k=2, and exactly for k>2.