Embedding spanning bounded degree graphs in randomly perturbed graphs

We study the model $G_\alpha\cup G(n,p)$ 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, which we call assisted 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 $\Delta\ge 5$, and every $n$-vertex graph $F$ with maximum degree at most $\Delta$, we show that if $p=\omega(n^{-2/(\Delta+1)})$ then $G_\alpha \cup G(n,p)$ 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 $G_\alpha \cup G(n,p)$ is lower than the appearance threshold in $G(n,p)$ by substantially more than a $\log$-factor. We prove that, for every $k\geq 2$ and $\alpha>0$, there is some $\eta>0$ for which the $k$th power of a Hamilton cycle with high probability appears in $G_\alpha \cup G(n,p)$ when $p=\omega(n^{-1/k-\eta})$. The appearance threshold of the $k$th 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 $k>2$.

§1. Introduction and results. Many 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. For example, Dirac's Theorem [13], one of the cornerstones of Extremal Graph Theory, states that an n-vertex graph with minimum degree at least n/2 has a Hamilton cycle when n 3. In Random Graph Theory, on the other hand, bounds are sought on the probability threshold for the appearance of subgraphs in a random graph. Let G(n, p) be the binomial random graph model with vertex set [n], where each possible edge is chosen independently at random with probability p. We say that G(n, p) has some property P with high probability (whp) if lim n→∞ P[G(n, p) ∈ P] = 1. A key result by Pósa [35] and Koršunov [26] is that G(n, p) with high probability contains a Hamilton cycle if p = ω(log n/n), whereas if p = o(log n/n), then G(n, p) with high probability does not. Here, we write p(n) = ω( f (n)) to signify p(n)/ f (n) → ∞, and p(n) = o( f (n)) to signify p(n)/ f (n) → 0.
The study of randomly perturbed graphs combines these two approaches by taking the union of a graph satisfying some minimum degree condition and a random graph G(n, p). The goal is then to determine which minimum degree conditions and edge probabilities suffice to guarantee some given subgraph with high probability. Bohman et al. [8], who pioneered the study of randomly perturbed graphs, proved that for every α > 0 the union of every n-vertex graph with minimum degree at least αn and a random graph G(n, p) with p = ω(1/n) contains whp a Hamilton cycle. This result shows that, compared to Dirac's Theorem, a much smaller minimum degree condition suffices in a randomly perturbed graph, and compared to the random graph G(n, p) alone a log-term improvement in the edge probability is possible.
The recent increased interest in randomly perturbed graphs sparked a collection of results of a similar flavour, typically featuring a small linear minimum degree condition and a log-term improvement in the edge probability. In this paper, we contribute to this body of research by developing a new general method for establishing such results for spanning subgraphs. Our approach uses an absorbing method. We show that this new approach gives simpler proofs of a number of known results, whose original proofs often use the regularity method and are therefore technically more complex. It also allows us to give strong new results concerning powers of Hamilton cycles and general bounded degree spanning subgraphs in randomly perturbed graphs. In particular, our result on powers of Hamilton cycles provides the first example for graphs with an n (1) improvement in the edge probability compared to G(n, p). A similar phenomenon was already discovered in the context of hypergraphs by McDowell and Mycroft [30], which we will return to in our concluding remarks.
Before discussing our techniques and results in more detail, we set our work in context by summarising related results in random graphs and randomly perturbed graphs. 1.1. Thresholds in G(n,p). We say that the functionp : N → [0, 1] is a threshold for a graph property P, if lim n→∞ P[G(n, p) ∈ P] = 0 whenever p = o(p), and 1 whenever p = ω(p) .
If only the latter is known to be true, then we say thatp is an upper bound for the threshold for P in G(n, p). Containing a graph as a (not necessarily induced) subgraph is a monotone property and therefore it has a threshold by a result of Bollobás and Thomason [10]. In the following, we will focus on spanning subgraphs. In their seminal work, Erdős and Rényi [15] proved that the threshold for perfect matchings in G(n, p) is log n/n. Pósa [35] and Koršunov [26] independently showed that the property of having a Hamilton cycle has the same threshold.
The problem of finding powers of Hamilton cycles as a subgraph is generally considered a stepping stone towards results for more general spanning subgraphs. The kth power G (k) of a graph G is the graph obtained from G by connecting all vertices at distance at most k. Kühn and Osthus [29] observed that the threshold in G(n, p) for the kth power of a Hamilton cycle when k 3 is n −1/k ; this follows from a general embedding theorem due to Riordan [36] (see Theorem 2.4). Similarly, the threshold of the square of a Hamilton cycle is conjectured to be n −1/2 , but this is still open. Currently, the best known upper bound, by Nenadov and Škorić [33], is off by a O(log 4 n)-factor from this conjectured threshold.
For a graph H, an H-factor on n vertices is the vertex disjoint union of copies of H with n vertices in total. An almost H-factor in an n-vertex graph G is a subgraph of G that is an H-factor on (1 − ε)n vertices. A breakthrough result was achieved by Johansson et al. [23] who showed that the threshold for a K +1 -factor, that is n +1 vertex-disjoint copies of K +1 , is given by p = log 1/ n n 2/( +1) .
In fact, their result concerns, more generally, H-factors for strictly balanced graphs H. The 1-density of a graph H on at least two vertices is and a graph is called strictly balanced if H is the only maximiser in m 1 (H ). Johansson et al. [23] proved that for factors of strictly balanced graphs H, the threshold is n −1/m 1 (H ) log 1/e(H ) n. Gerke and McDowell [20], on the other hand, showed that for certain (but not all) graphs H which are not strictly balanced, this threshold is n −1/m 1 (H ) . Let us now turn to larger classes of graphs. For bounded degree spanning trees, the second author [32] showed that, for each fixed , log n/n is the appearance threshold for single spanning trees with maximum degree at most (see also [31]).
More generally, let F (n, ) be the family of graphs on n vertices with maximum degree at most . For some constant C, Alon and Füredi [3] proved that, if p C(log n/n) 1/ , then G(n, p) contains any single graph from F (n, ) whp. This is far from optimal and, since the clique-factor is widely believed to have the highest appearance threshold among the graphs in F (n, ), the following well-known conjecture is natural. CONJECTURE 1.1. If ∈ N, F ∈ F (n, ) and p = ω(p ), then G(n, p) whp contains a copy of F .

For
= 2, this conjecture was very recently resolved by Ferber et al. [16], who in fact showed a stronger universality statement, where all graphs in F (n, ) are found simultaneously. For larger , Riordan [36] gave a general result (see Theorem 2.4), which requires an edge probability within a factor of n (1/ 2 ) from p . The current best result in the direction of Conjecture 1.1 is the following almost spanning version by Ferber et al. [17]. THEOREM 1.2 (Ferber et al. [17]). Let ε > 0 and 5. For every F ∈ F ((1 − ε)n, ) and p = ω(p ) the random graph G(n, p) whp contains a copy of F .
The approach in [17] is based on ideas from Conlon et al. [12], who proved a stronger universality statement for the almost spanning case while using the edge probability n −1/( −1) log 5 n. Theorem 1.2 for = 3 was thus already known (up to a log-factor), whereas the case for = 4 remains open. For spanning subgraphs, very recently, Ferber and Nenadov [18] showed that for p (log 3 n/n) 1/( −1/2) the random graph G(n, p) whp contains all graphs in F (n, ) universally.
In the almost spanning case, the log-term in p is expected to be redundant [17], but this remains open. In this paper, we will show that the log-term in p is redundant, even in the spanning case, if we add G(n, p) to a deterministic graph with linear minimum degree.
1.2. Randomly perturbed graphs. Bohman et al. [9] introduced the following model of randomly perturbed graphs. For α ∈ (0, 1) and an integer n, we first let G α be any n-vertex graph with minimum degree at least αn. We then reveal more edges among the vertices of G α independently at random with probability p. The resulting graph G α ∪ G(n, p) is a randomly perturbed graph and we are interested in its properties. In particular, research has focused on comparing thresholds in G α ∪ G(n, p) to thresholds in G(n, p).
Again, we concentrate on spanning subgraphs. Note that the existence of such subgraphs in G α ∪ G(n, p) is a monotone property (in G(n, p)), and thus has a threshold. Of course, if α 1/2, then G α is itself Hamiltonian by Dirac's Theorem. For α ∈ (0, 1/2), Bohman et al. [9] showed the existence of some c = c(α) > 0 so that, if p = c/n, then, for any G α , there is a Hamilton cycle in G α ∪ G(n, p) whp. They also proved that this is optimal: there exists some c > 0 so that there are graphs G α such that G α ∪ G(n, c /n) is not Hamiltonian whp. Comparing this threshold to the threshold for Hamiltonicity in G(n, p), we note an extra factor of log n in the latter. This log n term is necessary to guarantee minimum degree at least 2 in G(n, p)-otherwise clearly no Hamilton cycle exists. In the model G α ∪ G(n, p), however, this already holds in G α alone.
Krivelevich et al. [28] studied the corresponding problem for the containment of spanning trees of maximum degree in G α ∪ G(n, p). For p = c(ε, )/n, it is already possible to find any almost spanning bounded degree tree on (1 − ε)n vertices in G(n, p) [4]. The addition of G α then ensures there are no isolated vertices, and Krivelevich et al. [28] showed that this indeed allows every vertex to be incorporated into the embedding. They thus prove that, for α > 0, maximum degree and p = c(α, )/n, every spanning bounded degree tree is contained in G α ∪ G(n, p).
Very recently, Balogh et al. [5] determined the threshold of appearance for general factors in the model G α ∪ G(n, p). They proved that for every H, if p = ω(n −1/m 1 (H ) ), then G α ∪ G(n, p) contains an H-factor whp. Comparing this to the result of Johansson et al. [23], we observe again a saving of a log-term. For the graphs H covered by the result of Gerke and McDowell [20], on the other hand, we see that the thresholds in G α ∪ G(n, p) and in G(n, p) are the same.
Other monotone properties considered in the randomly perturbed graph model include containing a fixed sized clique, having small diameter, being k-connected [8] and being non-2-colourable [37]. 1.3. Our results. Our main contribution to the study of randomly perturbed graphs is the introduction of a new approach for obtaining results concerning spanning subgraphs. The basic idea is to use some random edges with the assistance of the deterministic edges to create so-called reservoir sets. Our key technical result is Theorem 3.3, which gives a condition for applying this method to spanning subgraphs. We defer the statement of this result along with the necessary definitions to § 3.
Using our method, we analyse the model G α ∪ G(n, p) with respect to the containment of spanning bounded degree graphs, addressing a problem which was highlighted by Krivelevich et al. in the concluding remarks of [28]. We obtain the following result.
5 be an integer and G α be a graph with minimum degree at least αn. Then, for every F ∈ F (n, ) and p = ω(n −2/( +1) ), whp G α ∪ G(n, p) contains a copy of F .
Our bound on p in Theorem 1.3 is best possible in the following sense. In the case where F is a K +1 -factor on n vertices and G α is a complete bipartite graph with parts of size αn and (1 − α)n, we need to find an almost spanning K +1 -factor on (1 − α( + 1))n vertices in G(n, p). This can easily be shown to require p = (n −2/( +1) ). Note in addition that the edge probability used in Theorem 1.3 is lower by a log-term in comparison to the anticipated threshold for the graph F to appear in G(n, p) (see Conjecture 1.1).
Our second result deals with powers of Hamilton cycles. Here we can save a polynomial factor n (1) compared to the threshold n −1/k in G(n, p). THEOREM 1.4. For each k 2 and α > 0, there is some η > 0, such that if G α is an n-vertex graph with minimum degree at least αn, then G α ∪ G(n, n −1/k−η ) whp contains the kth power of a Hamilton cycle.
It was proved by Komlós et al. [25] that G α on its own contains the kth power of a Hamilton cycle, provided that α k/(k + 1) and v(G α ) is large enough. Bedenknecht et al. [6] showed that for any k 3 there is an η so that G α ∪ G(n, n −1/k−η ) whp contains the kth power of a Hamilton cycle if α > c k for some absolute constant c k > 0.
Bennett et al. [7] provide the following lower bound. With G α the complete bipartite graph with αn and (1 − α)n vertices in the classes, they show that p has to be at least n −1/k(1−2α) for G α ∪ G(n, p) to contain the kth power of a Hamilton cycle. It would be interesting to determine the optimal dependence between α, k and η in Theorem 1.4.
Organisation. We finish this section by providing some further notation, before outlining our general embedding method for randomly perturbed graphs in § 3. We then prove Theorem 1.4, the less technical of our implementations of this method, in § 4. Theorem 1.3 is proved in § 5, with the proofs of two auxiliary lemmas given in § 6. Finally, we make some concluding remarks and sketch how our methods can give simpler proofs of other results in the literature concerning randomly perturbed graphs in § 7. Our results concern the embedding of certain graphs F in G α ∪ G(n, p). For obtaining such an embedding, our first step will always be to embed an almost spanning subgraph F * of F , and our second step then (working in an auxiliary graph on [2n] vertices) extends this to an embedding of F .
For the second step, we shall use the following hypergraph matching theorem of Aharoni and Haxell [1]. The setup will be as follows. F F * consists of t well-separated subgraphs S 1 , . . . , S t of F , and we shall encode all valid embeddings of S i that extend the embedding of F * as the edges of a hypergraph L i . The goal then is to find a hypergraph matching using exactly one edge from each L i . A hypergraph is r-uniform if each of its edges has cardinality r.
When we want to use this theorem, we need to verify the condition on L i . For this purpose, we shall use Janson's inequality (see, e.g., [ and any 0 < γ < 1, we have . This result will also be useful for the first step described above, in which we embed an almost spanning subgraph. In particular, the appearance of almost H-factors in G(n, p) for p Cn −1/m 1 (H ) is a straightforward consequence of Janson's inequality (see, e.g., [22,Theorem 4.9]). Here we need a minor modification of this result. For two graphs H 1 and H 2 , an (H 1 , H 2 )-factor is any graph that consists only of vertex disjoint copies of H 1 and H 2 . The following theorem concerning the appearance of an almost (H 1 , H 2 )-factors in G(n, p) can be proved with trivial modifications to the proof of [22,Theorem 4.9]. For our result on spanning bounded degree subgraphs, we shall also use the following result of Riordan [36], which allows the embedding of spanning subgraphs that are not locally too dense in G(n, p). For a graph H, let Riordan's Theorem can be found in the following form in [34]. We shall use this theorem for a subgraph H of F which excludes the "dense spots" of F . THEOREM 2.4 (Riordan [36]). Let 2 be an integer, H ∈ F (n, ) and p = ω(n − 1 γ (H ) ). Then, a copy of H is contained in G(n, p) whp.
Finally, we shall use the following submartingale-type inequality to handle weak dependencies in the proof of our main technical result. A proof of this lemma can, for example, be found in [ Main technical theorem. We start with an outline of the main idea of our strategy for embedding some spanning graph F into G α ∪ G(n, p). Recall that G(n, p) has vertex set [n]. We use two-round exposure. In the first round, we will find an F * -copy for some almost spanning induced subgraph F * of F . One key idea in our proof is that, by symmetry, the F * -copy we find is random among all possible F * -copies in the complete graph on vertex set [n] (see § 3.1). Hence, it remains to complete such a random F * -copy to an F -copy using only edges in G α and the second round (see § 3.2.). It is the additional edges of G α in this second step that allow us to gain on the bound for embedding F in a random graph alone.
For the second round, we use an absorbing method, relying on the following family of reservoir sets.
Definition 3.1 (Reservoir sets). Given a graph G α on vertex set [n], a copyF of a subgraph F * of F in the complete graph on vertex set [n] and an independent set W of vertices ofF , we define the family of (G α ,F , W )-reservoir sets(R(u)) u∈[n] by setting The crucial property of these reservoir sets is as follows. Assume thatF is a copy of F * in G(n, p). Then, for any vertex u ∈ [n] V (F ) exchanging u with any vertex w ∈ R(u) gives us a different copy of F * in G α ∪ G(n, p), now using u. In this case, we also say that we can switch u and w. Moreover, since W is an independent set, switching several vertices simultaneously in this manner does not create conflicts. As part of our proof we will show (see Lemma 3.5) that, for a randomF and a suitably chosen set W , the sets R(u) are likely to have linear size intersections with neighbourhoods in G α . This will give us "enough room" to completeF to F .
Next, we will state the technical embedding theorem, Theorem 3.3, that formalises this method. Theorems 1.4 and 1.3 will be inferred from this result. In our technical theorem, we are given, along with F , a family F of almost spanning subgraphs of F . This family is chosen such that whp one of these subgraphs appears in our first round and such that in our second round whp each subgraph in F can be extended to F , using vertex switching. We call a set F with these properties suitable, defined formally as follows.
each graph in F has at least (1 − ε)n vertices and the following two properties hold.
Observe that in (A2) we consider auxiliary graphs on [2n]. These encode all the information we need from G α and our second round of randomness. The sets B(v) then are the corresponding auxiliary versions of our reservoir sets. This setup, using [2n], allows us to keep the auxiliary reservoir sets disjoint from the F * -copy. The idea is, if F * can be extended to F in this auxiliary graph, then this corresponds to a homomorphism of F in the original setting on [n], and we can use switches to turn this homomorphism into an embedding.
We remark that in the proof of our first result, Theorem 1.4 on squares of Hamilton cycles, the family F only contains a single graph. In the proof of Theorem 1.3, however, the use of a larger family is crucial.

THEOREM 3.3 (Main technical result). Let α > 0 and ∈ N be constant and let p = p(n).
If G α and F are n-vertex graphs such that The main work for deducing our main results from this theorem will go into finding an (α, p)-suitable family F. Verifying (A1) corresponds to finding an almost spanning embedding for some F * ∈ F, which is usually not too hard, because εn vertices remain uncovered. To show (A2), by the definition of the B(v) there is a linear number of options for the embedding of every vertex, which makes this step again be somewhat similar to an almost spanning embedding (and we can also use the edges of G).
We will argue in § 7 that using this theorem we can also easily derive short proofs for a number of related results from the literature. We now turn to the proof of Theorem 3.3.

3.1.
Reducing the problem to completing a random subgraph copy. In this section we show that, using two-round exposure and (A1), we can reduce the problem of embedding F in G α ∪ G(n, p) to extending a random copy of an almost spanning subgraph. LEMMA 3.4. Let α, , p and G α , F, and F be as in the hypothesis of Theorem 3.3. For each F * ∈ F, letF be a random F * -copy in the complete graph on vertex set [n], and assume Then G α ∪ G(n, p) whp contains a copy of F .
Proof. Let G 1 and G 2 be two independent copies of G(n, p/2). For finding a copy of F in G(n, p), we want to use the edges of G 1 to find a copy of F * ∈ F, and then use (3) to complete such a copy to F using the edges of G 2 and G α . For the second step, we will condition on the success of the first step. For this purpose, we define the following events. Let F * 1 , . . . , F * r be the graphs in F. For each 1 i r, let E i be the event that there is a copy of F * i in G 1 , but no copy of F * j for every j < i. Note that this event is empty if F * j is a subgraph of F * i for some j < i. These events are chosen such that where the second equality uses (A1). In order to use (3) in the second step, it is essential that we obtain a random copy of F * ∈ F in the first step. Here, the crucial observation is that for each i ∈ [r] and a random F * i -copyF i in the complete graph on vertex set [n], we have Indeed, this follows from the fact that G 1 is independent of G α ∪ G 2 , and that, if we condition on E i , then G 1 contains an F * i -copy by definition and by symmetry each possible F * i -copy is equally likely to appear in G 1 . It follows that Completing a random subgraph copy. In this section, we provide the proof of our main technical theorem, Theorem 3.3. By Lemma 3.4, it remains to show that whp we can complete a random F * -copy into a copy of F . For this we will choose a large 2-independent set W in the F * -copy, which has no neighbours outside F * (this is with respect to F * as a subgraph of F ), construct the according reservoir sets and perform switches. Recall that a set W of vertices in a graph is called 2-independent, if it is independent and no pair of distinct vertices in W have a common neighbour. The following lemma, whose proof we defer to the end of the section, states that these reservoir sets are well distributed with respect to G α -neighbourhoods. LEMMA 3.5. Let α, , p and G α , F and F be as in Theorem 3.3. Let F * ∈ F and let W * be a maximally 2-independent set in F * , which has no neighbours outside F * . LetF be a random copy of F * in the complete graph on vertex set [n] and W be the image of W * inF .
This lemma in particular implies that the sets R(u) are linear in size.
Proof of Theorem 3.3. Assume that we are given graphs G α and F satisfying the assumptions and a suitable set of almost spanning subgraphs F of F . Fix F * ∈ F and letF be a random copy of F * in the complete graph on vertex set [n] and let g 0 be the embedding that maps F * toF .
By Lemma 3.4, it suffices to prove (3). For this purpose, we will use the reservoir sets and (A2). So, let W * be a maximally 2-independent set in F * , which has no neighbours outside We now start by mapping the remaining vertices of F arbitrarily to the unused vertices [n] V (F ). Our goal then is to use switchings to turn this mapping into an embedding of F . So, label the vertices in In order to appeal to (A2), we now define an auxiliary graph G on vertex set [2n] together with a collection of auxiliary reservoir sets B(u), which encode the embedding g 0 of F * and the edges of G α as well as the reservoir sets R(u).
Let G be the auxiliary graph on the vertex set [2n] that contains all edges ofF in addition to exactly the following edges. For each edge uw of G α , the graph G contains the edges {u + n, w}, {u, w + n} and {u + n, w + n}.
The corresponding embedding g of F into G ∪ G 2 extends g 0 . In particular, this F -copy in the auxiliary graph encodes which vertices get switched where (as we detail below). Now we need to translate this back to our original setting on n vertices. For this, let G 2 be the graph on vertex set [n] and with all edges uw such that {u, w + n}, {u + n, w} or {u + n, w + n} is an edge in G 2 . Hence, G 2 is distributed as a random graph in which each edge appears independently and with probability at most p/2. Therefore, in order to show (3), it is sufficient to prove that whenever the event E holds for G 2 , then there also is an F -copy in G α ∪F ∪ G 2 .
Indeed, assume that E holds and define for each v ∈ V (F ) otherwise .
In other words, the first line states that all vertices The third line guarantees that vertices v in V (F * ) usually are embedded by g as by g, unless this creates a conflict with the rule from the first line for a vertex u, in which case they are switched to z u by the second line. We claim that g is an embedding of F into G α ∪F ∪ G 2 . To see this, let Note that gagrees with g outside of Z 0 ∪ Z 1 , so that g(appropriately restricted) is an embedding Recall that R(z u ) ⊆ W by the definition of the reservoir sets, and W is the image under g of W * . We conclude that Z 1 ⊆ W * , that is, Z 1 is 2-independent and has no neighbours outside F * . It follows that vertices in where the last step uses g (v) ∈ R(z u ). This shows that vertices in Z 1 are properly embedded by g.
It remains to consider vertices v ∈ Z 0 . We prove that all neighbours of v are mapped to neighbours of g(v), distinguishing three cases. Firstly, for We conclude that g is an embedding of F into G α ∪F ∪ G 2 , completing the proof of Theorem 3.3.
It remains to prove Lemma 3.5, which is based on the fact that the reservoir sets R(u) are random sets.
Proof of Lemma 3.5. Note that, as F has maximum degree at most , we have Let g 0 be the (random) mapping of F * toF , and observe that, by symmetry, W = g 0 (W * ) is a uniformly random set of size |W * | in [n].
Fix u, v ∈ V (G α ). For each w * ∈ W * , note that |N F * (w * )| and that the sets {w * } ∪ N F * (w * ) are all disjoint. Let I w * be the indicator variable for the event g 0 (w * ) ∈ N G α (v) and Let r = αn 3 2 |W | and pick distinct vertices w * 1 , . . . , w * r in W * . Consider revealing the random copyF by, firstly, revealing the mapping of vertices in {w * 1 } ∪ N F * (w * 1 ), then revealing the mapping of vertices in {w * 2 } ∪ N F * (w * 2 ), and so on, until {w * r } ∪ N F * (w * r ), before finally revealing the rest of the vertices inF . Note that, for each 1 i r, when the location of the vertices in Therefore, by (6) and Lemma 2.5 applied with δ = ( α 4 ) +1 , we have . Using a union bound, we conclude that with probability Powers of Hamilton cycles. Let F = C (k) n be the kth power of the cycle with n vertices, and let P (k) n denote the kth power of a path with n vertices. To prove Theorem 1.4, it is sufficient, by Theorem 3.3, to find an η = η(α) > 0, such that there exists an (α, p)-suitable set F of subgraphs of F with In fact, we will use only one subgraph, which will consist of disjoint copies of the kth power of long (but constant length) paths, which we connect by shorter kth powers of paths to form a copy of F .
In the following, we shall explain how we choose F, and show that F satisfies (A1) and (A2) for p = n −1/k−η , which implies that F is (α, p)-suitable. We use the following constants. Given k and α > 0, let = 2k and ε = ( α 4 ) 2 . Pick large integers m and , and a small constant η > 0 such that where, for example, by 1 m η we mean that the following proof works if we choose η sufficiently small compared to 1/m. In particular, we require 2 εm. 4.1. Choosing F. Let F solely contain F * , the following (P (k) m , P (k) m+1 )-factor on at least (1 − ε)n vertices, which is a subgraph of F . Let s and t be the unique integers such that n = s(m + ) + t and t < (m + ). Let F * be the graph on v(F * ) = sm + t = t (m + 1) + (s − t )m vertices consisting of the following vertex disjoint kth powers of paths: t copies of P (k) m+1 , which we denote by P * 1 , . . . , P * t , and s − t copies of P (k) m , which we denote by P * t+1 , . . . , P * s . This leaves exactly v(F ) − v(F * ) = s sεm εn vertices of F uncovered. Observe that we obtain F from F * by connecting for each i ∈ [s] the paths P * i and P * i+1 (respectively, P * 1 if i = s) by a kth power of a path with vertices, which we denote by is the kth power of a path with + 2k vertices.

4.2.
Proof that F satisfies (A1). We use Theorem 2.3 to find a copy of F * in G(n, p/2). Since for m 2k, we have e(P (k) m ) = km − k+1 2 , it is easy to check that for k 2 we have )-factor on at most (1 − ε)n vertices, it follows directly from Theorem 2.3 that G(n, p/2) contains a copy of F * , and hence (A1) holds for F. 4.3. Proof that F satisfies (A2). Suppose that G is a graph with vertex set [2n] which contains a copyF of F * . For each v ∈ V (F ) V (F * ), assume we are given a set For each i ∈ [s] and j ∈ [k], let u i, j be the image of u * i, j inF , and v i, j be the image of v * i, j inF . Hence, to extendF to a copy of F we need to embed all vertices w * i, j with i ∈ [s] and j ∈ [ ] to distinct vertices w i, j so that is the kth power of a path with 2k + vertices. We would like to appeal to Hall's condition for hypergraphs, Theorem 2.1, to show that this is possible. For this purpose, we define the following auxiliary hypergraphs. Let W = [2n] V (F ). For each i ∈ [s], let L i be the -uniform hypergraph with vertex set W where e ∈ W is an edge exactly if there is some ordering of e as w i,1 , . . . , w i, so that (8) is the kth power of a path in G ∪ G and w i, j ∈ B(w * i, j ) for each j ∈ [ ]. We shall argue that the following lemma, whose proof we defer to § 4.4, guarantees that the assumption of Theorem 2.1 is satisfied.  , the hypergraph i∈A L i contains a matching with size greater than (|A| − 1). Indeed, let A ⊆ [s] and r = |A|, and let U be the vertex set of a maximal matching in i∈A L i . This means that there is no i ∈ A and edge e ∈ E (L i ) with V (e) ⊆ W U . Thus, by the property from Lemma 4.1, we have |U | 2 |A|, so that i∈A L i contains a matching with size at least |A|. Therefore, we can apply Theorem 2.1, and obtain a function π : with all the edges between the vertices u i removed and all the edges between the vertices v i removed. Furthermore, remove from P the edges u k w 1 , w v 1 and all the edges w i w i+1 , i ∈ [ − 1], except for w j w j+1 . The edges that we have removed will come from the deterministic graph G, while we will find a copy of P in G . The edge w j w j+1 is included in P so that we do not need to find a path between v k and w 1 in G.
To simplify our calculations for the application of Janson's inequality, let us first prove three simple claims concerning the density of subgraphs of P. Let U = {u 1 , . . . , u k } and V = {v 1 , . . . , v k }. Proof of Claim 4.2. In the ordering of the vertices in P in (9), ignoring the edge w j w j+1 , each vertex has at most k − 1 neighbours to the right. Therefore, including the edge w j w j+1 , we have e(P) ( + 2k)(k − 1) + 1 (k − 1/2), since we chose k. Proof of Claim 4.3. Removing the edge w j w j+1 if necessary, we have that each vertex in P has at most (k − 1) neighbours to the right in the labelling in (9). As the rightmost vertex in P has no such neighbours, if v(P ) 3, then we have e(P ) (k − 1)(v(P ) − 1) + 1 k(v(P ) − 1) − 1. If v(P ) = 2, then e(P ) = 1 k(v(P ) − 1) − 1. Therefore, as Proof of Claim 4.4. For such a subgraph P , let W 0 = V (P ) (U ∪ V ). We enumerate the vertices from W 0 by w i 1 , . . . , w i t from left to right in the ordering (9). If there is an index a with i a+1 − i a k + 1, then we estimate the number of edges in P through (k − 1)|W 0 | + 1. This is so because we can enumerate all the edges of P by identifying at least one vertex adjacent to every edge of P as follows: every vertex w i c (c a) is adjacent to the left to at most k − 1 vertices, and every vertex w i c (c > a) is adjacent to the right to at most k − 1 vertices, the only exception being possibly the vertices w j and w j+1 along with the edge w j w j+1 , thus contributing one more possible edge. Therefore, if |W 0 | 2, then If |W 0 | = 1, then, as k, the vertex in W 0 cannot have neighbours in both U and V , so that If there is no such index a as above, then note that v(P ) |W 0 | ( − k)/k k 2 . Then, counting from the edges of P from their leftmost vertex in (9), and remembering that w j w j+1 may be an edge, e(P ) Therefore, as η 1/ , For each i ∈ [A], let P i be the set of copies of P in the graph G with vertices in order (to match (9)) Note that, choosing the vertices in order w i,1 , . . . , w i, j , w i, , w i, −1 , . . . , w i, j+1 , there are at least 4εn − |U | − s 2εn options for each vertex, and therefore |P i | = (n ). Let P = ∪ i∈A P i , so that |P| = (r · n ).
For each Q, Q ∈ P, with Q = Q , let Q ∼ Q if Q and Q share some edge. Let q = p/6, the edge probability in G . Denote the expectation for the number of graphs from P in G by μ = |P|q e(P) and let δ = Q,Q ∈P: Q∼Q q 2e(P)−e(Q∩Q ) .
Note that, as η 1/k and p = n −1/k−η , we have q (k−1/2) n = ω(log n). As |P| = (r · n ), we then have, using Claim 4.2, Recall that for j ∈ [k] the vertices u i, j and v i, j denote the images of the end-k-tuples of the graph P * i from F * given through the copyF of F * and moreover that all these k-tuples contain distinct vertices (cf. § 4.3). Let i ∈ A and Q ∈ P i , and let U i = {u i,1 , . . . , u i,k } and Therefore, as |P| = O(rn ) and μ = (rq e(P) n ), using Claims 4.3 and 4.4. Thus, as μ = ω(r log n) and δ μ 2 = o(r −1 log −1 n), we can infer from Janson's inequality, Lemma 2.2, that with probability at least 1 − exp(−ω(r log n)) there is some i ∈ A and Q ∈ P i in G , and hence V (Q) (U i ∪ V i ) ∈ E (L i ), as required. §5. Spanning subgraphs with bounded maximum degree. Let F ∈ F (n, ) and p = ω(n − 2 +1 ). As before, we find a suitable set F of large subgraphs of F such that we can whp embed one of these subgraphs F * ∈ F in G(n, p/2) ((A1) in Definition 3.2), and then extend any such F * -copy (in an auxiliary graph) to cover all of F ((A2) in Definition 3.2). To do this, we adapt the strategy of Ferber et al. [17] to decompose F . In [17], each graph F ∈ F (n, ) is decomposed into a sparse part and many dense spots. Our set F will consist of subgraphs of F covering the sparse part and most of the dense parts.
Recall that the parameter determines when we can apply Riordan's theorem, Theorem 2.4, to embed a spanning subgraph in G(n, p). In the following, we call a graph H dense if γ (H ) > +1 2 and sparse otherwise. We can now define, following [17], a good decomposition of a graph.
Definition 5.1 (ε-good decomposition)). Let ε > 0, F ∈ F (n, ) and let S 1 , . . . , S k be families of induced subgraphs of F . For F = F − ( h S∈S h V (S)), we say that (F , S 1 , . . . , S k ) is an ε-good decomposition if the following hold. We remark that our definition is slightly less restrictive than that from [17], where (P3 ) is replaced by a stronger condition. An ε-good decomposition can easily be found using a greedy algorithm. The following lemma is proved in [17]. LEMMA 5.2 [17,Lemma 2.2]. For each ε > 0 and > 0, there exists some k 0 such that, for each F ∈ F (n, ), there is some k k 0 and an ε-good decomposition (F , S 1 , . . . , S k ) of F .
In the following, we shall use this lemma to define a family F of subgraphs of F ∈ F (n, ). We shall then show that this family F satisfies (A1) and (A2) and hence is (α, p)-suitable, which by Theorem 3.3 implies Theorem 1.3 as desired. 5.1. Choosing F. Fix F ∈ F (n, ). Let ε = ( α 4 ) 2 , and let k 0 be large enough for the result of Lemma 5.2 to hold with ε and . By Lemma 5.2, for some k k 0 , there is an ε-good decomposition (F , S 1 , . . . , S k ) of F , which we fix.
For each 1 h k, let s h be the size of the graphs in S h (possible by (P3)), and, picking some representative S ∈ S h , note that, by (P2) and as (S) , we have so that s h < 2 + 2. Thus, we may consider α, , ε, k k 0 and the maximum size of each dense spot (2 + 1) to be constant, while n tends to infinity. Let F contain exactly those induced subgraphs of F which cover F and, for each 1 h k, all but at most εn s 2 h k of the graphs from S h .

5.2.
Proof that F satisfies (A1). We shall embed the copy of F using Riordan's theorem, Theorem 2.4. In [17], the embedding of F is then extended step by step to include the graphs in S h , for 1 h k. We proceed similarly, but in each step only include most of the graphs S h , for 1 h k. This allows us to work at a lower probability than that used in [17], as we aim to find a copy of only some graph in F.
To find such a copy of a graph in F, we expose the graph G(n, p/2) in a total of k + 1 rounds, revealing G h ∼ G(n, q) for 0 h k, where q = p/(6k) and thus (1 − q) k+1 1 − p/2. Every edge is thus present with probability at most p/2 in h G h . We use G 0 to embed F and then iteratively use G 1 , . . . , G k to embed as many subgraphs from S 1 , . . . , S k as possible, and show that this results whp in an embedding of a subgraph from F.
Since, by (P1), γ (F ) +1 2 , and thus q = ω(n − 1 γ (F ) ), by Theorem 2.4, we can whp embed For 1 h k, we want to (whp) use edges from G h to extend the embedding f h−1 to cover all but at most εn s 2 h k graphs from S h . We then let f h be the extended embedding and let F h be the subgraph of F embedded by f h . We use the following lemma, which allows us to extend the current embedding to one more dense spot S ∈ S h , even if we restrict its image to a small but linearly sized set U , using only edges of G h . This lemma is proved along with another lemma from this section in § 6.
Start with f 0 and F 0 . For each 1 h k, we construct f h and F h , as follows. The property in Lemma 5.3 whp holds for h. We extend the embedding f h−1 to f h using edges from G h to cover as many of the graphs in S h as possible (with any edges to F h−1 correctly embedded), and call the resulting graph F h . By the property in Lemma 5.3, this leaves at most εn s 2 h k graphs in S h unembedded. Indeed, if there is a set S of at least εn s 2 h k unembedded graphs in S h , then, let U = V (G α ) V (F h ) and note that |U | s h · |S| εn s h k . There then exists some S ∈ S and a copy S of S in G h [U ] with isomorphism π : V (S) → V (S ) such that (10) holds for each v ∈ V (S). As, by (P5), no two subgraphs in S h have an edge between them, π can be used to embed S and extend the embedding f h , a contradiction.
From this, we obtain (whp) the embedding f k of a subgraph of F , covering F and all but at most εn s 2 h k graphs from each S h , 1 h k, into h G h . Such a subgraph embedded by f k is thus in F, and therefore (A1) holds. 5.3. Proof that F satisfies (A2). Let F * ∈ F and let the graph G be as described in (A2) containing the copyF of F * .
For each 1 h k, let S h ⊆ S h be those dense parts not in F * , so that |S h | εn s 2 h k . We have, for each 1 h k, that the graphs in S h are isomorphic, minimally dense, disjoint and neither have edges between them nor share any neighbours. Furthermore, the sets in Let G 1 , . . . , G k be independent random graphs with G i ∼ G(2n, q), where q = p/(6k). Starting with g 0 and F 0 = F * , for each 1 h k in turn, we will (whp) inductively find a function Note that g 0 satisfies these properties, and that, once we find g k whp, we will have an embedding of F k = F into G ∪ ( 1 h k G h ), satisfying the conditions in (A2). Noting that each edge in h G h appears independently at random with probability at most p/6, we then have that (A2) holds.
Suppose then that 1 h k and we have found the function g h−1 satisfying (Q1) and (Q2). S,1 , . . . , z S,s h }, and let L S be the s h -uniform auxiliary hypergraph with vertex set W h−1 , where e is an edge of L S if, for some labelling e = {w S,1 , . . . , w S,s h }, the map z S,i → w S,i is an embedding of S into G ∪ G h , where, for each 1 i s h we have w S,i ∈ B(z S,i ) and Each hyperedge e = {w S,1 , . . . , w S,s h } of L S then corresponds to a possible extension of g h−1 to cover S ∈ S h .
We wish to show that whp there exists a function π : S h → S∈S h E (L S ) such that π (S) ∈ E (L S ) for each S ∈ S h , and the edges in π (S h ) are pairwise vertex disjoint. This is possible, as shown below, using Theorem 2.1 and the following lemma. The property in Lemma 5.4 then holds for each 1 h k, 1 r |S h |, S ⊆ S h and U ⊆ W h−1 , with |S| = r and |U | s 2 h r with probability at least Similarly to our deductions from Lemma 4.1, it then follows that, for every S ⊆ S h , the hypergraph S∈S L S contains a matching with size greater than s h (|S| − 1). Therefore, by Theorem 2.1, a function π as described above exists. Thus, we can extend g h−1 to an embedding g h of F h satisfying (Q1) and (Q2) as required.
Subject to the proof of Lemma 5.4, this completes the proof that (A2) holds. §6. Proofs of auxiliary lemmas. In this section, we give the proofs of the lemmas from § 5.
Proof of Lemma 5.3. Fixing h, note that there are certainly at most 2 n · 2 n choices for S and U . Therefore, it is sufficient to prove, for fixed S ⊆ S h and Each graph in S is isomorphic to S 0 in F , but, when we come to extend an embedding of F h−1 to F h by embedding "most" of the copies from S h , the number of edges between a copy from S h and the already embedded F h−1 may differ. We now distinguish two cases: Case I where each copy S from S has some edge between S and F h−1 in F h and Case II where there is some copy S from S for which there is no such edge.
Let us assume first that we are in Case I. For each S ∈ S, let ) be the images of the already embedded neighbours of vertices in S. Note that these sets W S are nonempty by the definition of Case I and by (P5) are disjoint. For each H ∈ H and S ∈ S, let H ⊕ W S be the graph with vertex set V (H ) ∪ W S containing exactly those edges that we need in order to extend the partial embedding we have to embed S into H. That is, Note that, by (ii) of Claim 6.1, each graph in H + has at most ( + 1)s/2 edges. We will now consider a subfamily S of size at least 2|S| ( +1)s = (n) of those copies of S from S so that S ⊕ W S has the same number of edges, say m, where 1 m ( + 1)s/2. Using (ii) of Claim 6.1, and that q = ω(n − 2 +1 ), let the expected number of copies from H + S = {H ⊕ W S : Then, using (i) and (iii) of Claim 6.1, and as μ = (n s+1 q m ), we have Therefore, as μ = ω(n) and δ μ 2 = o(n −1 ), by Lemma 2.2, the probability that there is no graph in H + S in G h is at most exp(− μ 2 4(μ+δ) ) = exp(−ω(n)), as required. For Case I, it is left then only to prove Claim 6.1.
It remains to consider Case II. In this case, there is some graph from S ⊆ S h with no edges to F h−1 . Therefore, it is sufficient for some graph in H to exist. Let m = e(S 0 ) be the size of each (isomorphic) graph in H, and note that 2m min{s , s(s − 1)} (s − 1)( + 1). Thus, we may take Then, as μ = (n s q m ), we have Therefore, as μ = ω(n), and δ μ 2 = o(n −1 ), by Lemma 2.2, the probability that there is no graph in H in G h is at most exp(− μ 2 2(μ+δ) ) = exp(−ω(n)), as required. 6 and, in particular, Let s = s h . As in the proof of Lemma 5.3, we will consider two cases: Case I where each copy S from S has some edge between S and F h−1 in F h and Case II when for some copy S from S there is no such edge.
Suppose first that we are in Case I. For all S ∈ S, since (F ) and |S| = s, there are certainly at most s vertices in F h−1 with some edge in S, and at most 2 s ways of attaching such a vertex to S. Thus, we can consider a subfamily S of at least 1 2 s 2 |S| = (n) copies of S from S which are all isomorphic when the edges from S to F h are added to S. Pick S 0 ∈ S . Label V (S 0 ) = {v 1 , . . . , v s } so that v 1 has a neighbour in F h−1 . Recall that for S ∈ S we labelled V (S) = {z S,1 , . . . , z S,s h }. Without loss of generality, we can assume for each S ∈ S that v i → z S,i is an isomorphism from S 0 into S, and that z S,1 has a neighbour in F h in V (F h−1 ) (possible as we are in Case I). Let H be a set of |U | s copies of S 0 in the complete graph with vertex set U , where each copy of S 0 has a different vertex set. For each H ∈ H, label For each S ∈ S , pick the image w S of an already embedded neighbour of the vertex z S,1 corresponding to v 1 , that is, pick For each S ∈ S , note that, from (12), we have |H S | = (n s ).
) be the set of images of already embedded neighbours of vertices in S. For each H ∈ H S and S ∈ S , let H ⊕ W S be the graph with vertex set V (H ) ∪ W S and edge set These are exactly the edges we need in order to extend our embedding of and S ∈ S , E (H ⊕ W S ) does not include v H,1 w S , and, therefore, in place of (ii), the following holds. (ii') For each S ∈ S and H ∈ H S , 2e(H ⊕ W S ) ( + 1)s − 2.
Note that, by our choice of S , each graph in H + has the same number of edges, m say. Note that, as the property we are looking for is monotone, we may assume that q −1/2 = ω(log n).
We remark that this is the only place where we use that the edge v H,1 w S is not included in H ⊕ W S , since it is already present in G.
Defining δ as follows, and using similar deductions to those used to reach (11) Therefore, as μ = ω(r log n), and δ μ 2 = o(r −1 log −1 n), by Lemma 2.2, the probability that there is no graph in H + in G h is at most exp(− μ 2 2(μ+δ) ) = exp(−ω(r log n)), completing the proof of Lemma 5.4 in Case I.
Let us assume now we are in Case II, with some S 0 with no edges between S 0 ∈ S and Note that if we have some graph H ∈ H in G h , then we are done, as then V (H ) ∈ E (L S 0 ). From (12), we have |H| = (n s ), so, with very similar calculations to Case II in the proof of Lemma 5.3, we have that the probability that there exists no graph from H in G h is at most exp(−ω(n)) exp(−ω(r log(n/r)), as required. §7. Concluding remarks.
Extending Theorem 1.3 to smaller maximum degrees. Theorem 1.3 can be easily extended to 3 using basically the same approach as in § 5. The definition of the "dense spots," however, has to be slightly adapted to each case, but since it is straightforward, we omit the details. 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. This means that, using a similarly defined set of subgraphs F as in the proof of Theorem 1.3, we cannot show that one of these subgraphs appears whp in G(n, ω(n −2/5 )) (i.e., we cannot prove (A1) in Definition 3.2), and this prevents our approach from extending to this case.
Using our method. Our main technical theorem, Theorem 3.3, provides a new general purpose tool for finding spanning structures F in randomly perturbed graphs G α ∪ G(n, p). To use Theorem 3.3, it is sufficient to show that F has a collection of subgraphs which is (α, p)-suitable. Our approach avoids the regularity lemma, which appears in many previous proofs for results concerning spanning structures in G α ∪ G(n, p) [5,27,28]. In particular, our approach provides simpler proofs for recent results concerning bounded degree spanning trees and factors, as we sketch in the following.
Spanning trees. Krivelevich et al. [28] showed that, for any α, > 0, if p = ω(1/n) and T is an n-vertex tree with maximum degree at most , then G α ∪ G(n, p) contains a copy of T whp. We can reprove this result using Theorem 3.3 as follows. Fixing α > 0 and > 0, let ε = ε(α, ) be as given in Definition 3.2. Let p = ω(1/n) and let T be a tree with n vertices and maximum degree at most . Clearly T contains some subtree T with just over (1 − ε)n vertices, pick such a subtree and let F = {T }. By the work of Alon et al. [4], we know that G(n, p/2) whp contains a copy of T , and therefore (A1) holds for F. Furthermore, (A2) easily holds without even recourse to the random edges in G(2n, p/6). The copy of T can be extended by iteratively adding leaves. When we wish to add a leaf to a vertex w, say, to embed v ∈ V (F ) V (F ), as |B(v) ∩ N G (w)| 4εn, there will be many vertices to choose from in B(v) ∩ N G (w) which are not yet in the embedding. Thus, F is (α, p)-suitable and Theorem 3.3 applies.
Factors. Balogh et al. [5] showed that for every H, if p = ω(n −1/m 1 (H ) ), then G α ∪ G(n, p) contains an H-factor whp. Again, we can use Theorem 3.3 to easily reprove this result. Indeed, let F be an H-factor and F be the set of subgraphs of F consisting of disjoint copies of H which cover at least (1 − ε)n vertices. By Theorem 2.3, we have that (A1) holds for F. Another simple application of Janson's inequality gives that (A2) holds as well.
Randomly perturbed hypergraphs. Recently generalisations of the model of randomly perturbed graphs to hypergraphs attracted much attention. Again, the union of a binomial random r-uniform hypergraph G (r) (n, p) and a deterministic r-uniform hypergraph G α satisfying a certain minimum degree condition is considered. In the hypergraph setting, several different notions of minimum degree are possible.
The study of randomly perturbed hypergraphs was initiated by Krivelevich et al. [27] who considered hypergraphs G α with collective minimum degree αn, that is, each (r − 1)set of vertices of G α is contained in at least αn edges. A loose Hamilton cycle in an r-uniform hypergraph on n = (r − 1)k vertices for some integer k, is a labelling of its vertices by 0, . . . , n − 1 such that {i, . . . , i + (r − 1)} is an edge for each i = (r − 1) j with j ∈ {0, . . . , k − 1}, where indices are taken modulo n. In other words, consecutive edges of a loose Hamilton cycle overlap in exactly one vertex. We remark that, for loose Hamilton cycles, a Dirac-type theorem is known [24]. Krivelevich et al. [27] proved that, for any G α with collective minimum degree αn, the addition of random edges with edge probability c(α)n −r+1 (where c(α) > 0 depends on α only) is sufficient to create whp perfect matchings as well as loose Hamilton cycles. Comparing this to the threshold for matchings and loose Hamilton cycles in random hypergraphs, which is n −r+1 log n [14,19,23]), this again differs by a factor of log n.
Different minimum degree conditions were considered by McDowell and Mycroft [30]. An r-uniform hypergraph G α has minimum -degree at least αn r− if each -set of vertices of G α is contained in at least αn r− edges. An -overlapping cycle is defined analogously to a loose Hamilton cycle, but with consecutive edges overlapping in exactly vertices. A tight Hamilton cycle in an r-uniform hypergraph is an (r − 1)-overlapping Hamilton cycle. McDowell and Mycroft [30] showed that for -overlapping Hamilton cycles with 3 it is possible to save a polynomial factor n ε on the edge probability in randomly perturbed r-uniform hypergraphs G α ∪ G (r) (n, p) compared to G (r) (n, p) alone, under the assumption that G α has minimum -degree at least αn and minimum (r − )-degree at least αn r− . This result was extended by Bedenknecht et al. [6] to powers of tight Hamilton cycles, with the additional assumption of collective minimum degree at least αn with α > c r, . The weaker notion of minimum 1-degree was studied in the context of randomly perturbed hypergraphs by Han and Zhao [21]. It is not difficult to see that an r-uniform hypergraph with minimum collective degree at least αn has minimum 1-degree at least α n−1 r−1 . Hence, Han and Zhao [21] strengthen the results of Krivelevich,et al. by proving that adding c(α)n random edges to G α , whp creates a perfect matching and a loose Hamilton cycle. Furthermore, adding c(α)n r−1 random edges to G α gives rise to a tight Hamilton cycle. Both these results, as well as those from [27], use the regularity method.
The absorption technique we introduce in this paper can be extended to the randomly perturbed hypergraph model, and may allow some progress. In particular, we have confirmed that an easy extension of our method gives the appearance threshold for a perfect matching and a loose Hamilton cycle in this model, recovering the results of [21,27].
We note that the third and fourth of the current authors [34] have extended the result of Riordan [36] to hypergraphs. Similar extensions of Theorems 1.2 and 1.3 however remain open and would be very interesting.
Universality. We believe that a universality result corresponding to our main theorem holds as well. That is, we believe that when p = ω(n − 2 +1 ) the randomly perturbed graph G α ∪ G(n, p) contains whp a copy of every graph in F (n, ) simultaneously. However, our use of Riordan's result [36], which was proved by second moment calculations, makes it unlikely that our techniques can be used to obtain such a result. Thus, new ideas are required. Similarly, p is commonly believed to be the threshold for G(n, p) to contain a copy of every graph in F (n, ) simultaneously, but the current methods to attack this problem (see the discussion after Theorem 1.2) require an edge probability in distinct excess of this conjectured threshold.
In the case of spanning bounded degree trees, in joint work with Han and Kohayakawa, we establish the following universality result in [11]. We show that G α ∪ G(n, c(α, )/n) simultaneously contains all spanning trees of maximum degree at most . Mathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.