Embedding Spanning Bounded Degree Graphs in Randomly Perturbed Graphs

Abstract

We study the model G α ∪ G(n, p) of randomly perturbed dense graphs, where G α is any n-vertex graph with minimum degree at least αn 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 α > 0 and 5, and every n-vertex graph F with maximum degree at most , we show that if p = ω(n −2/( +1) ), then G α ∪ 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 α ∪ 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 α > 0, there is some η > 0 for which the kth power of a Hamilton cycle with high probability appears in G α ∪ G(n, p) when p = ω(n −1/k−η ). 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.

Universality for bounded degree spanning trees in randomly perturbed graphs

“…As mentioned earlier, the reservoir sets used in our proof are similar to those introduced in the setting of randomly perturbed graphs in . In that work, the reservoir sets are used to prove a general result about spanning structures in randomly perturbed graphs, which can be easily applied to consider the appearance of various different single spanning structures.…”

“…Theorem is an immediate consequence of a technical theorem, Theorem , which states that the union of G α with any reasonably expanding graph G is ‐universal. The proof of Theorem relies on the use of reservoir sets resembling those introduced in as part of the so‐called assisted absorption method. The novelty in our proof is that we construct these reservoir sets using expanding graphs rather than random graphs, which is not possible with the techniques from (see also the discussion in Section 2 and the proof of Lemma in Section 3.2).…”

“…The proof of Theorem relies on the use of reservoir sets resembling those introduced in as part of the so‐called assisted absorption method. The novelty in our proof is that we construct these reservoir sets using expanding graphs rather than random graphs, which is not possible with the techniques from (see also the discussion in Section 2 and the proof of Lemma in Section 3.2).…”

“…Before we turn to the details of our embedding technique, we mention further results concerning randomly perturbed graphs. Further spanning structures whose appearance in randomly perturbed graphs has been studied are F ‐factors (for fixed graphs F ) , squares of Hamilton cycles and copies of general bounded degree spanning graphs , perfect matchings and loose Hamilton cycles in uniform hypergraphs , and tight Hamilton cycles in hypergraphs . Most of the mentioned results exhibit the following phenomenon: in the presence of a dense graph G α , a smaller edge probability than in G ( n , p ) alone suffices.…”

Monochromatic Schur Triples in Randomly Perturbed Dense Sets of Integers

“…[6]. Further properties were studied in [4,17,25] and, even more recently, various spanning structures in randomly perturbed graphs and hypergraphs were investigated in [1,2,3,7,8,10,13,15,16,19].…”