Abstract
We study the model G 8 G(n; p) of randomly perturbed dense graphs, where G is any n-vertex graph with minimum degree at least 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 > 0 and C 5, and every n-vertex graph F with maximum degree at most , we show that if p = !(n−2~(+1)) then G 8 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 rst example of graphs where the appearance threshold in G 8 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 C 2 and > 0, there is some > 0 for which the kth power of a Hamilton cycle with high probability appears in G 8 G(n; p) when p = !(n−1~k−). 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.
Highlights
Introduction and resultsMany important results in Extremal Graph Theory and in Random Graph Theory concern the appearance of spanning subgraphs in dense graphs and in random graphs, respectively
In Extremal Graph Theory, minimum degree conditions forcing the appearance of such subgraphs are studied
There is no extension to = 4 of Theorem 1.2 due to the existence of one problematic dense spot: a triangle attached to the rest of the graph with two pendant edges at each vertex
Summary
2 we have m1(Pm(k)), m1(Pm(k+)1) < k. For each i ∈ [s], let Li be the -uniform hypergraph with vertex set W where e ∈ W is an edge exactly if there is some ordering of e as wi,1, . In the ordering of the vertices in P in (9), ignoring the edge w jw j+1, each vertex has at most k − 1 neighbours to the right. For each i ∈ [A], let Pi be the set of copies of P in the graph G with vertices in order (to match (9)). Let F contain exactly those induced subgraphs of F which cover F and, for each 1 h k, all but at most εn sh k of the graphs from. For 1 h k, we want to (whp) use edges from Gh to extend the embedding fh−1 to cover all most εn sh k graphs
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