Universality for bounded degree spanning trees in randomly perturbed graphs

We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph $G_\alpha$ on $n$ vertices with $\delta(G_\alpha)\ge \alpha n$ for $\alpha>0$ and we add to it the binomial random graph $G(n,C/n)$, then with high probability the graph $G_\alpha\cup G(n,C/n)$ contains copies of all spanning trees with maximum degree at most $\Delta$ simultaneously, where $C$ depends only on $\alpha$ and $\Delta$.

On the other hand, a large branch of the theory of random graphs studies when random graphs typically contain a copy of a given spanning structure F n . Let G(n, p) be the n-vertex binomial random graph, where each of the ( n 2 ) possible edges is present independently at random with probability p = p(n). A classical result of Bollobás and Thomason [8] states that every nontrivial monotone property has a threshold in G(n, p). Since containing a copy of (a sequence of graphs) F n is a monotone property, there exists a threshold functionp =p(n)∶ N → [0, 1] such that, if p = o(p), then lim n→∞ P[F n ⊆ G(n, p)] = 0, whereas, if p = (p), then lim n→∞ P[F n ⊆ G(n, p)] = 1. When the conclusion of the latter case holds, we say that G(n, p) contains F n asymptotically almost surely (a.a.s.). For example, a famous result of Koršunov [16] and Pósa [23] asserts that the threshold for Hamiltonicity in G(n, p) is (log n)∕n.
Bohman, Frieze, and Martin discovered the following phenomenon in [7]. Given a fixed > 0, they started with a graph G on n vertices with (G ) ≥ n. Here, can be arbitrarily small and hence G can be far from containing any Hamilton cycle. They proved that, after adding m = C( )n edges uniformly at random to G , the new graph G becomes Hamiltonian a.a.s., where C( ) is a constant that depends only on . Letting G be the complete unbalanced bipartite graph K n, (1− )n , one sees that the addition of linearly many edges to G is necessary for this result to hold in general. Furthermore, clearly, the conditions on (G ) and on p = m∕ ( n 2 ) in this result are weaker than in the corresponding Dirac-type problem and the threshold problem, respectively. More precisely, the probability p turns out to be smaller by a factor of Θ(log n). Here, we have switched from choosing m edges uniformly at random to the binomial G(n, p) model, which is known to be essentially equivalent when p = m∕ ( n 2 ) (see, e.g., [13]). The model G ∪ G(n, p) is known as the randomly perturbed graph model. Typically p = o(1), so an "addition" of G(n, p) to the dense graph G corresponds to a small random perturbation in the structure of G . This model and its related generalizations to hypergraphs and digraphs sparked a great deal of research in recent years.
In this paper we are concerned with spanning trees in randomly perturbed graphs. For almost spanning trees it was shown by Alon, Krivelevich and Sudakov [2] that, for some constant C = C( , Δ), the random graph G(n, C∕n) alone a.a.s. contains any tree with at most (1 − )n vertices and maximum degree at most Δ, where the bounds on C = C( , Δ) have subsequently been improved [3]. Since the random graph G(n, C∕n) a.a.s. contains isolated vertices, it obviously does not contain spanning trees. The problem of determining the threshold of bounded degree spanning trees attracted much attention. Recently, Montgomery [22] showed that for each constant Δ and every sequence of trees T n with maximum degree Δ, the threshold in G(n, p) for a copy of T n to appear is (log n)∕n (see also [21]). However, Krivelevich, Kwan and Sudakov [18] showed that, again, a smaller probability suffices in the randomly perturbed graph model. They proved that G ∪ G(n, p) a.a.s. contains a given spanning tree T n with maximum degree at most Δ when p = C(Δ, )∕n.
In the concluding remarks of [18], Krivelevich, Kwan and Sudakov raised the question of whether G ∪ G(n, D∕n) contains all spanning trees of maximum degree at most Δ simultaneously, for some constant D = D(Δ, ). The purpose of this paper is to answer their question in the affirmative. For stating our result we need some notation. For a family  of graphs, we say that a graph G is  -universal if G contains a copy of every graph F from  . We denote by  (n, Δ) the family of all trees of maximum degree at most Δ on n vertices. Theorem 1 For each > 0 and Δ ∈ N, there exists a constant D = D(Δ, ) such that the following holds. If G is an n-vertex graph with (G ) ≥ n, then the randomly perturbed graph G ∪ G(n, D∕n) is a.a.s.  (n, Δ)-universal.
This result is asymptotically optimal for 0 < < 1∕2, as with G the complete unbalanced bipartite graph K n,(1− )n we need a linear number of edges from G(n, p) already for the perfect matching. For > 1∕2 this follows from a more general result, applied to G alone, due to Komlós, Sárközy, and Szemerédi [15], for trees with maximum degree up to n∕ log n. Kim and Joos [14] have succeeded in transferring this result to the perturbed model. Theorem 1 is an immediate consequence of a technical theorem, Theorem 2, which states that the union of G with any reasonably expanding graph G is  (n, Δ)-universal. The proof of Theorem 2 relies on the use of reservoir sets resembling those introduced in [9] 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 [9] (see also the discussion in Section 2 and the proof of Lemma 5 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) [4], squares of Hamilton cycles and copies of general bounded degree spanning graphs [9], perfect matchings and loose Hamilton cycles in uniform hypergraphs [17], and tight Hamilton cycles in hypergraphs [11]. Most of the mentioned results exhibit the following phenomenon: in the presence of a dense graph G , a smaller edge probability than in G(n, p) alone suffices. The only exception to this rule so far are F-factors for certain nonstrictly balanced graphs F covered in [4]. Moreover, some variations of such results when is at least some positive constant c (which depends on other parameters of the problems at hand) were considered in [5,6,20].

NOTATION, MAIN TECHNICAL RESULT, AND PROOF OVERVIEW
We will use standard graph theoretic notation throughout. In the following, we briefly recap most of the relevant terminology. Given graphs G and H, write |G| = |V(G)| and G∖H = G[V(G)∖V(H)], that is, the induced subgraph of G on V(G)∖V(H). Throughout this note we omit floors and ceilings. For two not necessarily disjoint sets U and W of vertices of a graph G we write e(U, W) for the number of edges with one endpoint in U and the other in W, where we count edges that lie in U ∩ W twice. We say that an n-vertex graph G is an (n, p, , C)-graph if Δ(G) ≤ Cpn and, for any U, W ⊆ V(G) such that |U|, |W| ≥ n, we have e(U, W) ≥ (p∕C)|U||W|. We further denote the family of (n, p, , C)-graphs by (n, p, , C). Intuitively, the graphs from (n, p, , C) are graphs with a certain degree bound which are expanding for vertex subsets of linear size.
Our main technical result states that perturbing graphs G with minimum degree at least n by graphs G ∈ (n, D∕n, , C) results in  (n, Δ)-universal graphs.
Theorem 2 (Main technical result). For any > 0 and integers C ≥ 2 and Δ ≥ 1, there exist > 0, D 0 and n 0 such that the following holds for any D ≥ D 0 and n ≥ n 0 . Suppose G ∈ (n, D∕n, , C) and G are n-vertex graphs on the same vertex set and (G ) ≥ n. Then H ∶= G ∪G is  (n, Δ)-universal.
We will show in Section 5 that this result implies Theorem 1. In the remainder of this section, we give a brief outline of our proof of Theorem 2.

Proof overview
Let G ∈ (n, D∕n, , C). We embed an arbitrary T ∈  (n, Δ) into H ∶= G ∪ G in three phases. In the first phase, we find a subtree T 1 of T (see Lemma 3) of small linear size, say n with ≪ , and we embed this subtree T 1 into H using a randomized algorithm (see Lemma 5). In doing so, we can show that there is some such embedding in which, for any given pair of vertices -a fact which will turn out to be crucial later. We denote by B T,H (u, v) such a set of vertices w, and refer to such sets B T,H (u, v) as reservoir sets (see Section 3.2 for the formal definition). Alternatively, calling them switching sets would emphasize that each of them can only be used once.
In the second phase, we extend the tree T 1 to an almost spanning subtree T ′ of T with |T∖T ′ | = 2 n. For this purpose we use a theorem of Haxell [12] (see Corollary 6), which ensures such almost spanning embeddings exist given sufficient expansion in the host graph H.
Finally, in the third phase, we complete our embedding using a greedy approach and the reservoir sets B T,H (u, v) for the following swapping trick: since T ′ is a subtree of T, we can extend it by consecutively appending degree-1 vertices and thus growing the tree T is the sequence of subtrees of T that we encounter in this process. Suppose we already have the embedding , we proceed as follows. We will set things up so that, by counting, we will be able to show that there is some (We remark that we said that we would extend g i−1 to g i ; as it will be clear by now, this is not strictly speaking correct, as we may alter g i−1 slightly before extending it to g i .) As mentioned earlier, the reservoir sets used in our proof are similar to those introduced in the setting of randomly perturbed graphs in [9]. 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. In particular, this gives a short proof of the appearance of any single bounded degree spanning tree in this model, a problem that was first solved in [18]. The argument from [9] does not work for universality statements. However, here we show that the reservoirs can be found and the swapping trick employed in the completely deterministic setting by embedding the first part of the tree in a randomized way.

AUXILIARY LEMMAS
The lemmas provided in this section will be used in the proof of Theorem 2. We start in Section 3.1 with two lemmas for partitioning the tree T we want to embed. We then explain how we obtain good reservoir sets by embedding a subtree T 1 of T randomly in Section 3.2. Finally, in Section 3.3 we provide the tools to extend this embedding to an almost spanning subgraph of T.

Tree partitioning lemmas
Recall that  (n, Δ) is the collection of all trees on n vertices with maximum degree at most Δ, and that a graph G on n vertices is said to be  (n, Δ)-universal if G contains a copy of T for every T ∈  (n, Δ).
The main assertion of the following lemma is that we can find in any bounded degree tree T a subtree T 1 of roughly any desired size so that removing T 1 from T leaves a tree. We will use this lemma to find a small linear sized subtree T 1 , which we embed in our first phase.

Lemma 3
Let , > 0 and let n, Δ be positive integers such that Δ + 2 < 1. Then, for any Proof Fix any vertex v of T as the root and, for each w ∈ V(T), write C w for the branch (subtree) of T consisting of w and all of its descendants. By < 1∕Δ, deg(v) ≤ Δ and averaging, there is Note that (b) holds by the definition of T 1 . Finally, let T ′ be an (arbitrary) subtree of T such that T 1 ⊆ T ′ ⊆ T and |T∖T ′ | = 2 n. ▪ Let T be a tree. Given vertices x 1 , … , x m of T, let ⟨x 1 , … , x m ⟩ T be the minimal subtree of T that contains the vertices x 1 , … , x m , which is just the subtree of T obtained from the union of the vertex sets of all the paths between x i , x j , i ≠ j, in T. For two distinct vertices x, y of T, we write dist T (x, y) for their distance in T, namely, the length of the (unique) path on T connecting x and y. Given a vertex x of T and a vertex set The following lemma provides us with vertices x 1 , … , x s in a tree T which cover T well, but are not too close. In particular, this gives us a collection of stars x i ∪ N T (x i ) which are far enough apart that they are relatively independent.

A randomized embedding-controlling reservoir sets
In the following, we define formally the reservoir sets B T,H (u, v), already mentioned in the proof overview given in Section 2.1, and show that we can force them to be suitably large. These reservoir sets will be helpful when finishing the embedding of T, since they will allow us to alter locally partial embeddings that we construct sequentially. We warn the reader that, for technical convenience, the sets B T,H (u, v) are defined here in a slightly different manner in comparison with the informal definition given earlier in Section 2.1. Let V be a set of n vertices. Let G be a graph on V and let T be a tree with For distinct vertices u and v ∈ V, we define their reservoir set B T,G (u, v) as follows: Recall that the idea is that we can free up any w ∈ B T,G (u, v) used already in the embedding, by moving the vertex embedded to w to v. This then allows us to use w for embedding any unembedded neighbor of the vertex embedded to u. Our next lemma shows that we can embed the linear sized subtree T 1 of T into H = G ∪ G using a randomized algorithm, such that we get large reservoir sets.
Proof of Lemma 5 First we choose the parameters D 0 and as follows: and then we choose n 0 large enough. We apply Lemma 4 to T 1 and obtain s ∈ N and vertices x 1 , … , x s ∈ V(T 1 ) such that, for any Our embedding of T 1 consists of three steps. First we iteratively embed the disjoint stars with centers at x 1 , … , x s uniformly at random into stars in H (using only the edges of G) whose vertices have not yet been used as images. Next we connect these stars and obtain an embedding of a subtree of T 1 as the union of the stars and ⟨x 1 , … , x s ⟩ T 1 . At last we embed the rest of the vertices of T 1 greedily, which will be possible using G as |T 1 | ≤ n∕(2Δ) and (G ) ≥ n.
The following claim states that we can pick disjoint stars with Δ leaves (that is, copies of K 1,Δ ) in G, within which we will later embed the stars in T 1 with centers at x 1 , … , x s .

Claim
There is a choice of disjoint stars S 1 , … , S s with Δ leaves in G such that, for each u, v ∈ V there are at least 2(Δ + 3) n stars among S 1 , … , S s with their centers in N G (u) and their leaves in N G (v).
Proof of the Claim We randomly and sequentially pick s stars S 1 , … , S s with Δ leaves from G, where each star S i is picked uniformly at random from the copies of K 1,Δ which are disjoint from S 1 , … , S i−1 (we show below that this is indeed possible). For be the Bernoulli random variable for the event thatx i ∈ N G (u) and R i ⊆ N G (v), wherex i is the center of S i and R i is the set of leaves of S i . Since (G ) ≥ n, |T 1 | ≤ n∕(2Δ) and the existing stars cover at most vertices, there are at least 3 n∕4 vertices available in both . Since G ∈ (n, D∕n, , C) and 3 ∕4 ≥ , e(U, W) ≥ D|U||W|∕(Cn) ≥ 3 D|U|∕(4C). By the convexity of the binomial function, the number of K 1,Δ -stars with center in U and leaves in W is at least Since Δ(G) ≤ CD, the total number of K 1,Δ -stars in G is at most n ( CD . This allows us to obtain the following lower bound on by the choice of in (1). Thus, by Lemma 2.2 (the sequential dependence lemma) from [1] with = 1∕2, or a simple coupling argument, we get Thus by the union bound, we conclude that there is a choice of S 1 , … , S s such that, for each u, v ∈ V, Y u,v 1 + · · · + Y u,v s ≥ 2(Δ + 3) n, that is, the claim holds. ▪ Now let S 1 , … , S s be as given by the claim. Define the embedding g of the stars in T 1 on vertices Next we extend our forest by connecting these stars according to the order x 1 , … , x s , and obtain an embedding of a subtree of T 1 which is the union of the stars and ⟨x 1 , … , x s ⟩ T 1 . Suppose we have connected the first i − 1 stars, that is, we have an embedding of ⟨x 1 , … , x i−1 ⟩ T 1 , and now we will connect it tox i , the image of x i . Recall that dist T 1 (x i , ⟨x 1 , … , x i−1 ⟩ T 1 ) = 5 and thus let the path to be embedded be x i , y 1 , y 2 , y 3 , y 4 , z. Note that x i , z, y 1 are already embedded in H = G ∪ G . Moreover, if z ∈ {x 1 , … , x i−1 }, then y 4 has already been embedded; otherwise, fix a neighbor of g(z) in G which is not covered by the current partial forest as g(y 4 ). This is possible because (G ) ≥ n and |T 1 | ≤ n∕(2Δ). Note that, using G , there are at least n∕2 choices for the image of y 2 and at least n∕2 choices for the image of y 3 , so, as G ∈ (n, D∕n, , C), we can pickỹ 2 andỹ 3 so thatỹ 2ỹ3 is an edge of G. Thus, the sequencex i , g(y 1 ),ỹ 2 ,ỹ 3 , g(y 4 ), g(z) forms a path in H. Define g(y i ) =ỹ i for i = 2, 3. When finished, this completes the second step of the embedding.
For the last step, note that since the partial tree that has been embedded is connected, we can finish the embedding of T 1 by iteratively attaching leaves to the partial embedding. This is always possible because (G ) ≥ n and |T 1 | ≤ n∕(2Δ). Let g be the resulting embedding function andT 1 = g(T 1 ).
By the claim for any u, v ∈ V, there are at least 2(Δ + 3) n stars from S 1 , … , S s such that their centers are in N G (u) and their leaves are in N G (v). Since these stars are subtrees ofT 1 , we conclude that |BT 1 ,H (u, v)| ≥ 2(Δ + 3) n for any u, v ∈ V, as required. ▪

Almost spanning tree embeddings
To extend T 1 to the almost spanning tree T ′ , we will use the following corollary of a tree embedding result of Haxell [12] (this is her Theorem 1 with = 1 and each i = Δ). We note that it was first observed by Balogh, Csaba, Pei, and Samotij [3] that this is applicable in sparse random graphs. For a graph G and vertex set X ⊆ V(G), we let N G (X) ∶= ⋃ x∈X N G (x).

Corollary 6
Let T be a tree with t edges and maximum degree at most Δ. Suppose k ≥ 1 is an integer and G is a graph satisfying the following two conditions: Then G contains T as a subgraph. Moreover, for any vertex x 0 of T and any y ∈ V(G), there exists an embedding f of T into G such that f (x 0 ) = y.

MAIN TECHNICAL RESULT
In this section we prove our main technical result, Theorem 2. Given T ∈  (n, Δ) we will use Lemma 3 to obtain a subtree T 1 of T of small linear size, which we embed with the help of Lemma 5 and then extend to the embedding of an almost spanning subtree of T using Corollary 6. We then use the reservoir sets B T,H (u, v) to extend the embedding to cover the last few vertices.
Although we risk being somewhat repetitive, with the relevant definitions at hand, we are able to say more precisely how the sets B T,H (u, v) will help us to embed these last few vertices. Suppose we have a partial embedding g ∶ T ′ → H of our tree T into the host graph H, such that T ′ ⊆ T is connected and letT ′ = g(T ′ ). Since T ′ is a subtree in T we can extend it vertex by vertex by connecting T ′ with some new vertex b ∈ V(T∖T ′ ), which has one neighbor in V(T ′ ). Assume that this neighbor a of b in T has been embedded to u, but none of the unused vertices is connected to u in H so that we cannot simply embed b to one of the unused vertices. Instead, if there exists an unused vertex v such that BT ′ ,H (u, v) ≠ ∅, then we can proceed with the embedding as follows. Let w ∈ BT ′ ,H (u, v) and note that, by the definition of BT ′ ,H (u, v), we have w ∈ V(T ′ ). Let c = g −1 (w), and let g ′ (x) = g(x), for any x ∈ V(T ′ )∖{c}, g ′ (c) = v and g ′ (b) = w. Using the definition of BT ′ ,H (u, v), this gives a partial embedding g ′ into H with one more leaf, b, embedded. We will show that we only need this procedure to embed the last 2 n vertices of T, and, for any u, v ∈ V, by the property guaranteed by Lemma 5, the reservoir sets BT ′ ,H (u, v) will be large enough to proceed greedily.
For any u, v, w ∈ V and any two trees S and S ′ , observe that if N S (w) = N S ′ (w) and w ∈ B S,H (u, v), then w ∈ B S ′ ,H (u, v). Since, by construction, for any vertex w ∈ V(T 1 )∖{ã} we have NT 1 (w) = NT ′ (w), and so |BT ′ ,H (u, v)| ≥ |BT 1 ,H (u, v)| − 1 ≥ 2(Δ + 3) ′ n − 1 for any u, v ∈ V. It remains to embed the 2 ′ n vertices in V(T∖T ′ ) to H. We achieve this using BT ′ ,H (u, v) as explained at the beginning of this section. More precisely, since T ′ is connected, we can obtain T from T ′ by iteratively attaching one new leaf at a time, say using the sequence T ′ ∶= T ′ 0 ⊆ T ′ 1 ⊆ · · · ⊆ T ′ 2 n = T. We claim that we can extend the embedding inductively while keeping |BT′ i ,H (u, v)| ≥ |BT′ i−1 ,H (u, v)| − (Δ + 3) for every i ∈ [2 ′ n], where eachT ′ i is the image of T ′ i in H. Indeed, fix some index i ∈ [2 ′ n] and now we need to attach the vertex b i ∈ V(T ′ i ∖T ′ i−1 ), whose parent a i ∈ T ′ i−1 has been embedded toã i . Pick any vertex v ′ in V(H)∖V(T ′ i−1 ). Since we can pick w ∈ BT′ i−1 ,H (ã i , v ′ ) and let c = g −1 i−1 (w). We now swap c out of the current embedding and use its previous image w to embed b i , and embed c to v ′ instead. Precisely, define the new embedding g i by g i (x) = g i−1 (x) for any x ∈ V(T ′ i−1 )∖{c}, g i (c) = v ′ and g i (b i ) = w. LetT ′ i = g i (T ′ i ).

TREE UNIVERSALITY IN RANDOMLY PERTURBED DENSE GRAPHS
In this section, we show how Theorem 2 implies Theorem 1, using the following simple proposition.

Proposition 7
For any > 0 and C ≥ 2 there exists D 0 such that the following holds for any D ≥ D 0 . The random graph G(n, D∕n) a.a.s. contains some graph G ∈ (n, D∕n, , C).
Proof Choose D 0 such that D 0 ≥ 10 4 −2 . Let D ≥ D 0 and H ∶= G(n, D∕n). Note that, by a simple Chernoff bound, the probability that, for all U, W ⊆ V(H), with |U|, |W| ≥ n∕10, we have is at least 1 − 2 2n e −D 2 n∕4800 = 1 − o (1). Assume then that the property in (2) holds. We will show that there are few vertices with high degree in H. Let A ⊆ V(H) be the set of vertices with degree exceeding 5D∕4 in H, and note that it satisfies e H (A, V(H)) > 5D|A|∕4. Thus, by the property in (2), we have that |A| < n∕10.
If we delete all the edges incident to vertices of degree larger than CD ≥ 5D∕4 from H then we are left with a graph G of maximum degree at most CD satisfying that for any two sets U and W of size at least n, we have Thus, G is in (n, D∕n, , C), as required. ▪