The embedding

Decomposition of bounded degree graphs

As outlined above we will adapt much of the embedding strategy of Ferber, Luh, and Nguyen [54].

We therefore briefly sketch their approach here, while mentioning the tools we need. In [54], each graph F ∈ F(n,∆) is decomposed into a sparse part and many dense spots. For this recall that γ(H) := max{e(H⁰)/(v(H⁰)−2)) :H⁰ ⊆Handv(H⁰)≥3}, and call a graphGdenseifγ(H)> ^∆+1₂ andsparseotherwise.

Say that two graphsS, S⁰ ⊆F areisomorphic inF, if there exist labellingsV(S) ={v1, . . . , v_s}and V(S⁰) ={v⁰₁, . . . , v⁰_s}, such thatvj7→v_j⁰ is an isomorphism betweenSandS⁰and, for each1≤j≤r,

|N_F(v_j)\V(S)|=|N_F(v⁰_j)\V(S⁰)|.

Definition 5.1(ε-good decomposition). Letε >0,F ∈ F(n,∆)and letS1, . . . ,Skbe families of induced subgraphs ofF. ForF⁰=F−(S

h

S

S∈S_hV(S))we say that(F⁰,S1, . . . ,Sk)is anε-good decomposition if the following hold.

P1 F⁰is sparse, that is,γ(F⁰)≤^∆+1₂ . P2 EachS ∈ S

hSh is minimally dense, that is,γ(S) > ^∆+1₂ and S⁰ is sparse for allS⁰ ⊆S with 3 ≤ v(S⁰)< v(S).

P3 For each1≤h≤k, all the graphs inShare isomorphic inF. P4 EveryShcontains graphs on at mostεnvertices, that is|S

S∈ShV(S)| ≤εn. P5 All the graphs inS

iSiare vertex disjoint and, for each1≤h≤kandS, S⁰ ∈ ShwithS 6=S⁰, there are no edges betweenSandS⁰inF, andSandS⁰share no neighbours inF.

We call the graphs inS1, . . . ,Skthedense spotsof the decomposition.

Anε-good decomposition can easily be found using a greedy algorithm. The following lemma is proved in [54].

Lemma 5.2(Lemma 2.2 in [54]). For each ε > 0 and ∆ > 0, there exists somek₀ such that, for each F∈ F(n,∆), there is somek≤k0and anε-good decomposition(F⁰,S1, . . . ,Sk)ofF.

The sparse partF⁰can be embedded using Theorem 2.1, the result of Riordan [97]. The embedding ofF⁰is then extended in [54] step by step to include the graphs inSh, for1≤h≤k. This can be done using Janson’s inequality (Theorem 2.18) and a hypergraph analog of Hall’s theorem due to Aharoni and Haxell [2] (Theorem 2.20). These are the tools used to prove Theorem 2.4. To construct our reservoir structure in the second step of our proof, we will also need the concentration inequalities Theorem 2.16 and 2.17.

5.2 The embedding

Letα > 0and∆ ≥ 5, and takeε= (_4∆^α)^2∆. Letk0 be large enough for the result of Lemma 5.2 to hold withεand∆. LetF ∈ F(n,∆), and, for somek≤k0, using the property from Lemma 5.2, let (F⁰,S1, . . . ,Sk)be anε-good decomposition ofF. For each1≤h≤k, lets_hbe the size of the graphs 54

5. Randomly perturbed graphs

inSh(possible byP3), and, picking some representativeS ∈ Sh, note that, byP2and as∆(S)≤∆, we have

(∆ + 1)(s_h−2)<2e(S)≤∆s_h,

so thatsh<2∆ + 2. Thus, we may considerα,∆,ε,k≤k0, and the maximum size of each dense spot (2∆ + 1) to be constant, whilentends to infinity.

We will expose the graphG(n, p)in a total of2k+ 1rounds, revealingGh =G(n, q)for0≤h≤k and G⁰_h = G(n, q)for 1 ≤ h ≤ k, where(1−q)^2k+1 = 1−pand thusq = Θ(p). Every edge is thus present with probabilitypin(∪hGh)∪(∪hG⁰_h). We useG0, . . . , Gk to embed most ofF while embedding all ofF⁰, and then useG⁰₁, . . . , G⁰_kto finish the embedding. We let

G:=[

h

G_h.

Embedding most of the graph

Our goal here is to embed all but at mostεnvertices of the graphF while ensuring we embed all the vertices inF⁰. Since, byP1,γ(F⁰)≤ ^∆+1₂ , and thusq =ω(n^−γ(F10)), by Theorem 2.1, we can almost surely embedF⁰intoG₀. Letf₀:V(F⁰)→V(G₀)be such an embedding and letF₀⁰ =F⁰.

For1 ≤ h ≤ k, we want to (almost surely) use edges fromGh to extend the embeddingf_h−1to cover all but at most _s^εn2

hk graphs fromSh. We then letf_hbe the extended embedding and letF_h⁰ be the subgraph ofF embedded byfh. We use the following lemma, which allows us to extend the current embedding to one more dense spotS ∈ S_h, even if we restrict its image to small but linearly sized setU, using only random edges ofGh. This lemma is proved along with the other lemmas from this section in Section 5.3.

Lemma 5.3. For each1≤h≤k, the following holds a.a.s. for anyS ⊆ S_handU ⊆V(G_α)with|S| ≥ _s^εn2 hk

and|U| ≥ _s^εn

hk. There is someS∈ Sand a copyS⁰ofSinGh[U]with an embeddingπ:V(S)→V(S⁰)such that, for eachv∈V(S),

f_h−1(N_F(v)∩V(F_h−1⁰ ))⊆N_Gh(π(v)). (5.1)

So, we start withf0andF₀⁰. For each1≤h≤k, we constructfhandF_h⁰, as follows. The property in Lemma 5.3 almost surely holds forh. We extend the embeddingf_h−1tofhusing edges fromGhto cover as many of the graphs inShas possible (with any edges toF_h−1⁰ correctly embedded), and call the resulting graphF_h⁰. By Lemma 5.3 this leaves at most _s^εn2

hk graphs inShunembedded. Indeed, if there is a setSof at least _s^εn2

hk unembedded graphs inS_h, then, letU =V(G_α)\V(F_h⁰)and note that

|U| ≥sh· |S| ≥ _s^εn

hk. There then exists someS ∈ Sand a copyS⁰ofS inGh[U]with isomorphism π:V(S)→V(S⁰)such that (5.1) holds for eachv∈V(S). As, byP5, no two subgraphs inShhave an edge between them,πcan be used to embedSand extend the embeddingfh, a contradiction.

From this we (almost surely) obtain the embeddingf_kof a subgraph ofF, coveringF⁰and all but at most_s^εn2

hk graphs from eachSh,1≤h≤k, intoG=S

hGh.

5.2 The embedding

Independence of f

_k

and G

_α

Until now, we have only used random edges for our embedding, and, therefore, we can consider this embedding to be independent of the edges ofGα, which we make precise as follows. LetFbe the set of possible induced subgraphs ofF which coverF⁰and, for each1≤h≤k, all but at most_s^εn2

hk of the graphs fromSh. Letrbe the number of graph isomorphism classes inF, and letF₁^∗, . . . , F_r^∗ ∈ Fbe representatives of each class. In Section 5.2 we proved thatP[∃iwith some copy ofF_i^∗inG] = 1−o(1). For each1 ≤i ≤ r, letEi be the event that there is a copy ofF_i^∗inG, but no copy ofF_j^∗ for any probability that there exists andiwith some copy ofF_i^∗inG. We will show in Section??, for each 1≤i≤r, that

Hence, it remains to show (5.2). Before we can turn to this, we first need to prepare our reservoir structure. dense, disjoint, neither have edges between them nor share any neighbours. Furthermore, the sets in {V(F^∗)} ∪ {V(S) :S∈ S_h⁰,1≤h≤k}form a partition ofV(F). Note that|V(F)\V(F^∗)| ≤εn.

LetV0⊆V(F^∗)be a maximal independent set inF^∗of vertices with no neighbours inV(F)\V(F^∗) inF. Note that|V0| ≥(|F^∗| −∆|V(F)\V(F^∗)|)/(∆ + 1)≥n/2∆and letW0:=g0(V0)⊆V( ˆF). For each vertexv∈V(G_α), letB(v)⊆W₀be the set of verticesw∈W₀such that every neighbour ofwin 56

5. Randomly perturbed graphs

Fˆis also a neighbour ofvinGα. That is,

B(v) ={w∈W0:NFˆ(w)⊂NG_α(v)}. (5.3) We shall later, in the proof of Lemma 5.4, show that eachB(v)is a random set, which entails that it has properties convenient for our embedding strategy. Crucially, by the definition ofB(v)we can switchany vertex fromB(v)inFˆwithvand still get a copy ofF^∗. Hence, these sets form ourreservoir structure. Furthermore, the setsB(v)all lie inW0, a large independent set inFˆ, hence their preim-ages need no further neighbours added to extend the embeddingg₀ toF. Therefore, we can switch different vertices in this manner without creating conflicts. The following lemma states that almost surelyuhas linearly manyG_α-neighbours inB(v)for eachu, v ∈V(G_α). This, in particular, implies that almost surely for eachv∈V(Gα)the setB(v)is linear in size.

Lemma 5.4. A.a.s., for eachu, v∈V(Gα)we have|NG_α(u)∩B(v)| ≥4εn.

Again, we defer the proof of Lemma 5.4 to Section 5.3. We note that it is not difficult, but constitutes a crucial new idea of our proof. It relies on the fact that the setsB(v)are random sets.

Let us now briefly indicate how we will use the reservoir structure in the next subsection to finish the embedding. Essentially, we pair each vertexw ∈ V(F)\V(F^∗)with a different vertex,vwsay, inV(Gα)\V( ˆF). Then, we ‘embed’ eachw∈V(F)\V(F^∗)to some vertexzw ∈B(vw), where it is important here thatB(vw)has linear size. The only problem is thatzwalready has a vertex embedded to it, but by switchingz_wout ofFˆand replacing it withv_w, we can shift part of the original copy ofF^∗ to fix this. If we ensure that the verticeszwwithw∈V(F)\V(F^∗)are distinct then these switchings can be carried out simultaneously, completing the embedding.

Finishing the embedding

We want to show thatP[∃a copy ofF inGα∪Fˆ∪(S

hG⁰_h)] = 1−o(1), which is precisely (5.2) and completes the proof of Theorem 2.7. We will follow the approach of Ferber, Luh, and Nguyen [54] and use our reservoir structure as outlined above. Roughly speaking, in comparison to [54], the minimum degree condition into the setsB(v)guaranteed by Lemma 5.4 gives us one edge betweenFˆand each dense spotfor free, allowing the use of a lower edge probability in our result.

So, letF₀=F^∗and recall that we have an embeddingg₀:F₀→G_α∪G. Now, for each0≤h≤k letFh=F[V(F^∗)∪(S

h⁰≤h

S

S∈S_h0V(S))]. Noting that|V(Gα)\V( ˆF)|=|S

h

S

S∈S⁰_hV(S)|, label the vertices inV(G_α)\V( ˆF)as{v_S,i: 1≤h≤k, S∈ S_h⁰,1≤i≤s_h}.

Starting withg0, for each1≤h≤kin turn, we will (almost surely) find a function gh:V(Fh)→V( ˆF)∪ {vS,i: 1≤h⁰≤h, S ∈ Sh⁰,1≤i≤sh} such that

Q1 ghis an embedding ofFhintoGα∪G∪(S

h⁰≤hG⁰_h0), and

Q2 for each vertexvofF_h−1, except for at most ^εhn_k vertices inV₀, we haveg_h(v) =g_h−1(v).

5.2 The embedding

Note thatg0satisfies these properties, and that, once we a.a.s. findgk, we will have an embedding of F=FkintoGα∪G∪(S

hG⁰_h), as required.

Suppose then that1≤h≤kand we have already found the functiong_h−1. Then we define the set W_h−1={g0(v) :g_h−1(v) =g0(v), v∈V0}, that is, the vertices ofFˆinW0that have not been switched.

Note that|W₀\W_h−1| ≤ ^ε(h−1)n_k byQ2. For eachS ∈ S_h⁰, labelV(S) ={z_S,1, . . . , z_S,sh}, and letL_S be thesh-uniform auxiliary hypergraph with vertex setWh−1, whereeis an edge ofLS if, for some labellinge ={wS,1, . . . , wS,s_h}, the mapzS,i 7→ wS,iis an embedding ofS intoGα∪G⁰_h, where, for for eachS∈ S_h⁰, and the edges inπ(S_h⁰)are pairwise vertex disjoint. This is possible, as shown below, using Theorem 2.20 and the following lemma.

Lemma 5.5. For each 1 ≤ h≤ k,1 ≤ r ≤ |S_h⁰|,S ⊆ S_h⁰ andU ⊆ W_h−1, with|S| = rand |U| ≤s²_hr, the following holds with probability at least1−exp(−ω(rln(ⁿ_r))). There exists someS ∈ S and an edge e∈E(LS)withV(e)⊆W_h−1\U.

contains a matching with size at leasts_h|S|. Therefore, we can apply Theorem 2.20, and conclude that a functionπas described above exists.

5. Randomly perturbed graphs

We claim thatghis an embedding ofFhintoGα∪G∪(S

h⁰≤hG⁰_h0), so that alsoQ1holds. Let Z0:={v:v=zS,iorgh−1(v) =wS,ifor someS∈ S_h⁰,1≤i≤sh}.

Note thatgh agrees withg_h−1 outside ofZ0, so thatgh (appropriately restricted) is an embedding ofF_h−Z₀. By the definition ofZ₀ andP5, the only edges in F_h[Z₀]are those within eachS ∈ S_h⁰. For eachS ∈ S_h⁰ we have thatzS,i 7→ wS,i is an embedding of S intoGα∪G⁰_h. It follows thatgh

(appropriately restricted) is an embedding ofFh[Z0]. It remains only to check that the edges between Z₀andV(F_h)\Z₀are appropriately embedded byg_h. That is, we wish to show for eachv∈Z₀that

g_h(N_Fh(v)\Z₀)⊂N_Gα∪G∪(^S_{h0 ≤hG}⁰

h0)(g_h(v)).

We consider two cases. Firstly, for eachv = zS,iwith S ∈ Sh and1 ≤ i ≤ shthe vertexv has no neighbours inV0, and hence

gh(NF_h(v)\Z0) =gh(NF_h(zS,i)\Z0)=^Q2gh−1(NF_h(zS,i)\Z0)

(5.4)

⊆NG_α∪G⁰_h(wS,i) =NG_α∪G⁰_h(gh(v)). Secondly, for eachv∈Z0withgh−1(v) =wS,ifor someS∈ S_h⁰ and1≤i≤sh, we havegh(v) =wS,i. Moreover, by the choice ofV0we haveNF_h(v)⊂V(F0). ByQ2it follows that

g_h(N_Fh(v)\Z₀) =g₀(N_Fh(v)) =NFˆ(w_S,i)⊂N_Gα(v_S,i) =N_Gα(g(v)), where we have used the definition ofB(v)in (5.3) and thatw_S,i∈B(v_S,i)by (5.4).

Thus, we can a.a.s. extend the embeddingg₀to an embeddingg_kofF inG_α∪G∪(S

hG⁰_h), com-pleting the proof of the theorem.

Im Dokument Spanning structures in random graphs and hypergraphs (Seite 62-67)