Main lemmas and outline of the proof

Before we state the main lemmas that we use in the proof of Theorem 3.3 and outline roughly how they will be combined, we need to introduce some more definitions.

Let r, k ≥ 1 and let B_r^k be the graph on kr vertices obtained from a path on r vertices by replacing every vertex by a clique of size k and by replacing every edge by a complete bipartite graph minus a perfect matching. More precisely, we define B_r^k as

V(B_r^k) := [r]×[k]

and for all distinct j, j⁰ ∈[k]

{(i, j),(i⁰, j⁰)} ∈E(B_r^k) if and only if i=i⁰ or|i−i⁰|= 1.

We call B_r^k backbone graph on the vertex set [r]×[k].

Let K_r^k ⊆ B_r^k be the spanning subgraph of B^k_r that is the disjoint union of r complete graphs on k vertices given by the following components: the clique K_r^k[{(i,1), . . . ,(i, k)}] is the i-th component of K_r^k for each i ∈ [r]. See Figure 3.1 for an illustration of the clique factor K_r^k in a backbone graphB_r^k fork= 3.

Figure 3.1: Three components of the clique factor K_r³ in the backbone graphB³_r. A partitionV⁰ ={V_i,j}i∈[r],j∈[k] is called k-equitable if|V_i,j| − |V_i,j0|≤1 for everyi∈[r]

andj, j⁰ ∈[k]. Similarly, an integer partition{n_i,j}i∈[r],j∈[k]ofn(meaning thatn_i,j ∈Z_≥0 for everyi∈[r], j ∈[k] and P

i∈[r]j∈[k]ni,j =n) isk-equitable if|ni,j−n_i,j0| ≤1 for everyi∈[r]

andj, j⁰ ∈[k].

Now we are ready to sketch the idea of the proof of Theorem 3.3 and state the main lemmas that we need. The goal of the proof is to find a.a.s. for every subgraphGof G(n, p) with minimum degree at least (k−1)/k+γ

pna graph homomorphism from a given graph H, with the properties as in the statement of the theorem, to G.

The first main lemma (Lemma 3.4) is used to partitionG. It states that a.a.s. Γ =G(n, p) satisfies the following property ifp= Ω(logn/n)^1/2. For any spanning subgraphG⊆Γ with minimum degree a sufficiently large fraction of pn, there exists an (ε, d, p)_G-regular vertex partition V of V(G) whose reduced graph R^k_r contains a clique factor K_r^k, on which the corresponding vertex sets ofV are pairwise (ε, d, p)_G-super-regular. Furthermore, (G,V) has

one-sided and two-sided inheritance with respect to R^k_r, and the Γ-neighbourhoods of all vertices but the ones in the exceptional set of V have almost exactly their expected size in each cluster. The proof of Lemma 3.4 is given in Subsection 3.1.3.

Lemma 3.4(Lemma for G). For each γ >0, k≥2, andr₀≥1 there existsd >0 such that for everyε∈ 0,1/(2k)

there exist r₁ ≥1 andC^∗>0such that the following holds a.a.s. for Γ =G(n, p) if p≥C^∗(logn/n)^1/2.

Let G= (V, E) be a spanning subgraph of Γ with δ(G) ≥ (k−1)/k+γ

pn. Then there exists an integerr with r₀ ≤kr≤r₁, a subset V₀ ⊆V with |V₀| ≤C^∗max{p⁻², p⁻¹logn}, a k-equitable vertex partitionV ={V_i,j}i∈[r],j∈[k]of V(G)\V₀, and a graph R^k_r on the vertex set [r]×[k]with K_r^k⊆B_r^k⊆R^k_r, with δ(R_r^k)≥ (k−1)/k+γ/2

kr, and such that the following is true:

(G1) _4krⁿ ≤ |Vi,j| ≤ ⁴ⁿ_kr for every i∈[r]and j∈[k],

(G2) V is(ε, d, p)_G-regular on R_r^k and (ε, d, p)_G-super-regular on K_r^k, (G3) both NΓ(v, Vi,j), V_i0,j⁰

and NΓ(v⁰, Vi,j), NΓ(v, V_i0,j⁰)

are(ε, d, p)G-regular for every {(i, j),(i⁰, j⁰)} ∈E(R^k_r) and v∈V \V₀,

(G4) |N_Γ(v, V_i,j)|= (1±ε)p|V_i,j|for every i∈[r], j∈[k] and every v∈V \V₀. Furthermore, if we replace (G3) by

(G3’) N_Γ(v, V_i,j), V_i0,j⁰

is (ε, d, p)_G-regular for every {(i, j),(i⁰, j⁰)} ∈ E(R^k_r) and every v∈V \V₀,

then we have the stronger bound |V₀| ≤C^∗p⁻¹logn.

After Lemma 3.4 has constructed a regular partitionV of V(G), the second main lemma deals with the graph H that we would like to find as a subgraph ofG.

More precisely, Lemma 3.5 provides a homomorphismf from the graphHto the reduced graph R^k_r given by Lemma 3.4, which has among others the following properties. The edges of H are mapped to the edges of R^k_r, and the vast majority of the edges of H are assigned to edges of the clique factor K_r^k ⊆R^k_r. The number of vertices of H mapped to a vertex of R^k_r only differs slightly from the size of the corresponding cluster of V. The lemma further guarantees that each of the first √

βn vertices of the bandwidth ordering ofV(H) is mapped to (1, j) withj being the colour that the vertex has received by the given colouring of H.

In caseH is D-degenerate the next lemma also ensures that for every (i, j)∈[r]×[k], a constant fraction of vertices mapped to (i, j) have each at most 2Dneighbours.

Lemma 3.5(Lemma forH). GivenD, k, r≥1andξ, β >0the following holds ifξ ≤1/(kr) and β ≤10⁻¹⁰ξ²/(Dk⁴r).

Let H be a D-degenerate graph on n vertices, let L be a labelling of its vertex set of bandwidth at most βn and let σ : V(H) → {0, . . . k} be a proper (k+ 1)-colouring that is (10/ξ, β)-zero-free with respect to L, where the colour zero does not appear in the first √

βn vertices ofL. Furthermore, let R^k_r be a graph on vertex set[r]×[k]with K_r^k⊆B^k_r ⊆R^k_r such that for every i ∈[r] there exists a vertex z_i ∈ [r]\ {i}

×[k] with

z_i,(i, j) ∈E(R^k_r) for every j∈[k].

Then, given a k-equitable integer partition {m_i,j}i∈[r],j∈[k] of n with n/(10kr) ≤ m_i,j ≤ 10n/(kr) for everyi∈[r] and j∈[k], there exists a mapping f: V(H)→ [r]×[k] and a set of special vertices X⊆V(H) such that for every i∈[r] andj ∈[k]we have

3.1. The bandwidth theorem in random graphs 49

(H1) m_i,j−ξn≤ |f⁻¹(i, j)| ≤m_i,j+ξn, (H2) |X| ≤ξn,

(H3) {f(x), f(y)} ∈E(R^k_r) for every{x, y} ∈E(H), (H4) y, z ∈S

j⁰∈[k]f⁻¹(i, j⁰) for everyx∈f⁻¹(i, j)\X and {x, y},{y, z} ∈E(H), (H5) f(x) = 1, σ(x)

for all vertices x in the first √

βn vertices ofL, and (H6) |{x∈f⁻¹(i, j) : deg(x)≤2D}| ≥ _24D¹ |f⁻¹(i, j)|.

Lemma 3.5 is a strengthened version of [42, Lemma 8]. The proof of [42, Lemma 8] is deterministic; here we use a probabilistic argument to show the existence of a functionf that also satisfies the additional property (H6), which is required for the proof of Theorem 3.15, that we give in Section 3.2. However, we still borrow ideas from the proof of [42, Lemma 8].

The proof of Lemma 3.5 is presented in Subsection 3.1.4.

So far, the vertices of the exceptional set V₀ of the regular partition V of V(G) were disregarded. To cover them, we need to manually pre-embed vertices of Honto all vertices in V0. For this, we use vertices in H that are not in triangles, that are pairwise far apart from each other and that are contained in the firstβnvertices of the bandwidth orderingLofV(H).

Once we embedded a vertexxofHonto a vertexvofV₀, we also embed its neighboursN_H(x).

This creates restrictions on the vertices of Gto which we can embed the second neighbours, and for the application of the sparse blow-up lemma (Theorem 2.9) we need certain conditions to be satisfied. The next lemma states that we can find vertices in N_G(v) satisfying these conditions with room to spare. We prove Lemma 3.6 in Subsection 3.1.5.

Lemma 3.6 (Common neighbourhood lemma). For each d > 0, k ≥ 2, and ∆ ≥ 2 there exists α >0 such that for every ε^∗ ∈(0,1)there exists ε₀ >0 such that for every r ≥1 and every 0 < ε ≤ ε0 there exists C^∗ > 0 such that Γ = G(n, p) a.a.s. satisfies the following if p≥C^∗(logn/n)^1/∆.

Let G= (V, E) be a (not necessarily spanning) subgraph ofΓ and {V_i\W}i∈[k]∪ {W} a vertex partition of a subset of V such that the following is true for every distinct i, i⁰ ∈[k]:

(V1) _4krⁿ ≤ |Vi| ≤ ⁴ⁿ_kr,

(V2) (Vi, V_i0) is (ε, d, p)G-regular, (V3) |W|= 10^−10ε_k⁴4^pnr⁴, and

(V4) |N_G(w, V_i)| ≥dp|V_i|for everyw∈W. Then there exists a tuple (w₁, . . . , w_∆) ∈ ^W_∆

such that for every Λ,Λ^∗ ⊆ [∆], and every distincti, i⁰∈[k] we have

(W1) |T

j∈ΛN_G(w_j, V_i)| ≥αp^|Λ||V_i|, (W2) |T

j∈ΛN_Γ(w_j)| ≤(1 +ε^∗)p^|Λ|n, (W3) |T

j∈ΛN_Γ(w_j, V_i)|= (1±ε^∗)p^|Λ||V_i|, and (W4) T

j∈ΛNΓ(wj, Vi),T

j^∗∈Λ^∗NΓ(wj^∗, V_i0)

is (ε^∗, d, p)G-regular if |Λ|,|Λ^∗|<∆and ei-ther Λ∩Λ^∗ =∅ or ∆≥3 or both.

Let H⁰ and G⁰ denote the subgraphs of H and G that result from removing all vertices that were used in the pre-embedding process. As a last step before finally applying the sparse blow-up lemma, the clusters in V

G⁰ need to be adjusted to the sizes of W_i,j

H⁰. The next lemma states that this is possible, and that after this redistribution the regularity properties that we need for the sparse blow-up lemma (Theorem 2.9) hold. The proof of Lemma 3.7 is given in Subsection 3.1.6.

Lemma 3.7 (Balancing lemma). For all integers k, r₁,∆ ≥ 1, and reals γ, d > 0 and 0 <

ε <min{d,1/(2k)} there exist 0< ξ <1/(10kr₁) and C^∗ >0 such that the following is true for everyp≥C^∗(logn/n)^1/2 and every 10γ⁻¹ ≤r ≤r₁ provided that n is large enough.

LetΓbe a graph on the vertex set[n]and letG= (V, E)⊆Γbe a (not necessarily spanning) subgraph with vertex partitionV={V_i,j}i∈[r],j∈[k] that satisfiesn/(8kr)≤ |V_i,j| ≤4n/(kr) for eachi∈[r], j ∈[k]. Let {n_i,j}i∈[r],j∈[k] be an integer partition of P

i∈[r],j∈[k]|V_i,j|. Let R^k_r be a graph on the vertex set [r]×[k] with minimum degree δ(R^k_r) ≥ (k−1)/k+γ/2

kr such thatK_r^k ⊆B_r^k ⊆R^k_r. Suppose that the partition V satisfies the following properties for each i∈[r], all distinct j, j⁰∈[k], and eachv∈V:

(B1) n_i,j−ξn≤ |V_i,j| ≤n_i,j+ξn, (B2) V is ₄^ε, d, p

G-regular on R^k_r and ^ε₄, d, p

G-super-regular on K_r^k, (B3) both N_Γ(v, V_i,j), V_i,j0

and N_Γ(v, V_i,j), N_Γ(v, V_i,j0)

are ^ε₄, d, p

G-regular pairs, and (B4) |NΓ(v, Vi,j)|= 1±^ε₄

p|Vi,j|.

Then, there exists a partitionV⁰ ={V_i,j⁰ }i∈[r],j∈[k] of V such that the following properties hold for each i∈[r], all distinct j, j⁰∈[k], and eachv∈V:

(B1’) |V_i,j⁰ |=n_i,j,

(B2’) |V_i,j4V_i,j⁰ | ≤10⁻¹⁰ε⁴k⁻²r₁⁻²n,

(B3’) V⁰ is(ε, d, p)_G-regular onR^k_r and (ε, d, p)_G-super-regular on K_r^k, (B4’) both N_Γ(v, V_i,j⁰ ), V_i,j⁰ ₀

and N_Γ(v, V_i,j⁰ ), N_Γ(v, V_i,j⁰ ₀)

are (ε, d, p)_G-regular pairs, and (B5’) for each1≤s≤∆and every collection of svertices v1, . . . , vs∈[n]we have

N_Γ(v₁, . . . , v_s;V_i,j)4N_Γ(v₁, . . . , v_s;V_i,j⁰ )≤10⁻¹⁰ε⁴k⁻²r⁻²₁ deg_Γ(v₁, . . . , v_s) +C^∗logn .

Furthermore, if for any two disjoint vertex setsA, A⁰⊆V(Γ)with|A|,|A⁰| ≥ _50000kr¹ ₁ε²ξpnwe haveeΓ(A, A⁰)≤ 1 +₁₀₀¹ ε²ξ

p|A||A⁰|, and if ‘regular’ is replaced with ‘fully-regular’ in (B2), and (B3), then we can replace ‘regular’ with ‘fully-regular’ in (B3’) and (B4’).

The last step of the proof of Theorem 3.3 is the application of the sparse blow-up lemma (Theorem 2.9) to the vertex partition ofG⁰ given by Lemma 3.7 and to the vertex partition of H given by Lemma 3.5 restricted toH⁰ while respecting the image restrictions that resulted from the pre-embedding process.

Before we turn to the details of the proof of Theorem 3.3 in Subsection 3.1.7, we prove Lemmas 3.4–3.7 in Subsections 3.1.3–3.1.6. First of all we show in the next subsection an almost sure property ofG(n, p) that we need at numerous places in the proofs of this chapter.

3.1. The bandwidth theorem in random graphs 51

Im Dokument Existence and Enumeration of Spanning Structures in Sparse Graphs and Hypergraphs (Seite 57-61)