Theorem 2.7 can be shown to extend to∆ ≤3using basically the same approach as presented here.
The definition of thedense spots, however, has to be slightly adapted to each case. For∆ = 4, there is one problematic dense spot, a triangle attached to the rest of the graph with two pendant edges at each vertex, which prevents our methods extending to this case. A similar gap exists for extensions of Theorem 2.4.
With simple modifications, our methods are applicable more generally to the determination of appearance thresholds for spanning structures in the randomly perturbed graph model. This allows us to give simpler, non-regularity, proofs of results already found in the literature. In particular, we can reprove the recent results concerning bounded degree spanning trees and factors. We will show how our methods imply these results in the following. As the calculations and arguments are very similar to our proof, we shall be brief.
Spanning trees
Theorem 5.7(Krivelevich, Kwan and Sudakov [82]). For anyα,∆ > 0, ifp = ω(1/n)and T is an n-vertex tree with maximum degree at most∆, thenGα∪ G(n, p)contains a copy ofT a.a.s.
Applying our new approach to the randomly perturbed graph model, we can give a proof of this result, as follows.
Proof. Fixingα > 0and ∆ > 0, letε = ε(α,∆) be a small constant. Letp = ω(n−1)and letT be a tree withn vertices and maximum degree at most∆. Clearly T contains some subtreeT0 with d(1−ε)nevertices. By the work of Alon, Krivelevich and Sudakov[14], we know thatG(n, p)almost surely contains a copyS0ofT0. As before, we observe that this copy is independent ofGαand placed uniformly at random on top of it. Using the same methods as we used for Lemma 5.4, we obtain that, a.a.s. each uncovered vertexvinGαcan be switched for a setB(v)of vertices in the copy ofT0so that each vertex inGαhas at least(∆ + 1)εnneighbours inGαinB(v).
We then greedily extend the copy ofT0 to a copy ofT using the following deterministic strategy.
Picking an uncovered vertexv, letBvbe obtained fromB(v)by removing those vertices which have been switched inS0 or whose neighbours inS0 have been switched and note that we removed less than(∆ + 1)εnvertices. Picking a vertexuinT which needs to be embedded as a leaf of the partial embedding, we use that its already embedded parent has at least one neighbourwinBv. We switchv withwand, by embeddinguontow, gain an extended partial embedding ofT. When complete, this gives the required copy ofT.
Factors
Recall that by Theorem 2.2 n−1/d1(G)log1/e(G)ngives the threshold for factors of strictly balanced graphsG. Gerke and McDowell [61], on the other hand, showed that for vertex balanced graphsG, this threshold isn−1/m1(G). The result for general factors in the modelGα∪ G(n, p)is the following.
Theorem 5.8(Balogh, Treglown and Wagner [19]). For everyG, ifp=ω(n−1/m1(G)), thenGα∪ G(n, p) contains aG-factor a.a.s.
5.4 Proofs for trees and factors
Our methods give a simpler, non-regularity, proof of this result, as follows.
Proof. FixingG, lett =v(G),ε=ε(α, t)>0be small andp=ω(n−1/m1(G)). It follows from Theo-rem 2.19 thatG(n, p)a.a.s. contains an almostG-factor covering at least(1−εt)nvertices, which we callF∗. As before, we observe that this copy is independent ofGαand placed uniformly at random on top of it. Using the very same methods we used for Lemma 5.4, we get that, a.a.s. each uncovered vertexvinV(Gα)can be switched for one of at least3εt3nvertices in different copies ofGinF∗to get the same number of disjoint copies ofGinGα∪ G(n, p). We call this set of switchable verticesB(v).
We can then iteratively extend our embedding as follows. We picktuncovered verticesv1, . . . , vtin V(Gα)and pick disjoint setsBi⊂B(vi)with|Bi| ≥εnso that the vertices inBiare switchable with vi, the vertices inS
iBi appear in different copies ofG, and these copies ofGhave had no vertices switched with them. This is possible as there are at most εn copies ofG added, which together switched at mostεtnvertices and thus blocking at mostεt2nvertices. Furthermore, the previously chosenBi can block at most anotherεt2nmany vertices and since at most tvertices appear in the same copy ofGthis leaves us withεt2nswitchable vertices.
It easily follows from the proof of Theorem 2.19 that, for anytdisjoint vertex subsets with at least εnvertices in each subset, with probability at least1−n−2there is a copy ofGinG(n, p)with one vertex in each subset. ThereforeG(n, p)contains a copy ofGwith one vertex in each setBiand thus we can use this copy, along with switchings, to increase the number of disjoint copies ofG. As there are at mostεnsteps until completion, this process finds aG-factor inGα∪ G(n, p)a.a.s.
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Chapter 6
Universality in random hypergraphs
Next, we present the proof33of Theorem 2.9 onF(r)(n,∆)-universality inH(r)(n, p)obtained together with Person [94]. After the proof outline and some more definitions we state two lemmas which read-ily imply the theorem. The first lemma guarantees pseudorandom properties inH(r)(n, p), whereas the second shows that these are sufficient for embedding any graph fromF(r)(n,∆). Afterwards, we prove both lemmas.
6.1 Proof outline
Our proof follows a similar strategy as the one of Dellamonica, Kohayakawa, R ¨odl, and Ruci ´nski [41]
for universality of random graphs which we combine with the approach of Kim and Lee [73] and of Ferber, Nenadov and Peter [56].
We will embed any bounded degree hypergraphF ∈ F(r)(n,∆)into the random hypergraphH= H(r)(n, p)withp=C(logn/n)1/∆by verifying certain deterministic pseudorandom properties. More precisely, we introduce the notion of an(n, r, p, t, ε,∆)-good hypergraphH(see Definition 6.2 below), and prove that the random hypergraphH(r)(n, p)is(n, r, p, t, ε,∆)-good a.a.s. This reduces our task to find an embedding of anyF∈ F(r)(n,∆)into such an(n, r, p, t, ε,∆)-good hypergraphH.
Roughly speaking, such good hypergraph H admits a partition of its vertices into setsV0, V1, . . . , Vt, so that certainextension properties hold. Next we partition most of the vertices of F into 3-independent setsX1,. . . ,Xtplus an additional setX0=NF(Xt)(recall, that a set is3-independent if any two of its vertices are at distance at least4). These3-independent setsX1,. . . ,Xtare constructed by colouring greedily the third power of the shadow graph ofF. The setXthas the property that the(r−1)-uniform link hypergraph of every x ∈ Xt inF looks the same together with possibly some further edges ofFcontained inNF(x). Thus, we think ofF[X0]as a collection of vertex-disjoint copies of isomorphic pairs(E1, E2), which we call profiles, of the edge setE1of somer-uniform hy-pergraph and of the edge setE2 of some(r−1)-uniform hypergraph (isomorphic to every link of x∈Xt). One of the properties ofH asserts then thatF[X0]can be embedded intoH[V0], no matter whichF∈ F(r)(n,∆)we consider. Then we extend, using other properties of an(n, r, p, t, ε,∆)-good hypergraphH, our embedding introunds to the wholeV(F)by embedding in the i-th round the vertices fromXi intoV0∪V1. . .∪Vi. For this we will verify Hall’s condition for the existence of a matching in an appropriately defined bipartite graph that allows us to carry on with our embedding.
The role of the setsViis technical – it allows us to verify Hall’s condition for small subsets, which we cannot simply do inV0.
33The proof in this chapter is a close adaption of [94].