The lemma for G

This subsection is devoted to the proof of the lemma forG (Lemma 3.4). We borrow ideas from the proof of [41, Proposition 17] and from the proof of [39, Lemma 9].

Our proof strategy can be summarised as follows. First we apply Lemma 2.6 to obtain an equitable partition of V(G) within whose reduced graph we can find a backbone graph by Theorem 1.3. Then, starting with an empty setZ₁ =∅, we add every vertex v to Z₁ for which there exists a clusterU such that the size of the Γ-neighbourhood of vinU is not close top|U|, or for which the Γ-neihbourhood in the exceptional set is too large, or for which its Γ-neighbourhoods fail to inherit regularity. We also add a minimum number of extra vertices to maintaink-equitability and then remove temporarily the setZ₁ from the graph.

Now we consider each vertex that destroyed super-regularity on the clique factor of the backbone graph before the removal of Z₁. We redistribute these vertices as well as the vertices from the exceptional set of the partition to other clusters such that they do not destruct super-regularity anymore. The moving of the vertices may have destroyed some of the regularity inheritance, Γ-neighbourhood, and super-regularity properties we tried to obtain before. However, a vertex only witnesses failure of these properties if exceptionally many of its Γ-neighbours were moved from or to a cluster. We defineZ2 to be the set of all such vertices plus a minimum number of additional vertices to obtain k-equitability of the remaining partition, and removeZ₂ from the graph. We will see that Z₂ is so small that its removal does not significantly affect the desired properties. Hence we can setV₀ = Z₁∪Z₂ and have found a partition ofV(G) with the properties as demanded.

Proof of Lemma 3.4. First we fix the constants that we need in the proof. Given γ > 0, k≥2, and r₀ ≥1, setd=γ/32. Letβ >0 andn₀≥1 be returned by Theorem 1.3 for input γ/2, 3k, and k. Let

r⁰₀ = maxn

r₀, n₀,^k_d,^10k_β o .

Next, let ε₁ > 0 and C₁ > 0, and ε₂ > 0 and C₂ > 0 be returned by Lemma 2.10 and Lemma 2.11, respectively, each with inputε/2 and d. Furthermore, let C₃ >0 be returned by Proposition 3.8 with inputε^∗2/(1000k²). Givenε∈ 0,1/(2k)

, set ε^∗ = minn

ε1, ε2,₁₀^ε10^2γk²

o .

Then we apply Lemma 2.6 with input ε^∗/k, (k−1)/k+γ, and r₀⁰ +k in order to obtain r1 >0. Finally, we setC = max_i∈[3]{Ci} and

C^∗ = ^100k_ε²_∗^r31C. Givenp≥C^{∗ log}_nⁿ1/2

, it holds thatG(n, p) a.a.s. satisfies the good events of Lemmas 2.10 and 2.11, and of Proposition 3.8 with the parameters specified above. We condition on Γ =G(n, p) satisfying these good events.

Given G= (V, E) ⊆Γ with δ(G)≥ ^k−1_k +γ

pn, we apply Lemma 2.6, with input ε^∗/k, α= (k−1)/k+γ,r⁰₀+k, andd, to G. Observe that this is possible becauseGis a subgraph of Γ, and we have Cp⁻¹logn≤ε^∗n/(kr₁), so that the condition of Lemma 2.6 is satisfied as we have conditioned on the good event of Proposition 3.8 for Γ. The result is an (ε^∗/k, d, p)-regular partition U⁰ ofV with t⁰ ∈[r⁰₀+k, r₁] equally sized clusters, with exceptional set U₀⁰

3.1. The bandwidth theorem in random graphs 53

of size at mostε^∗n/kand whose reduced graph R has minimum degree at least δ(R)≥ ^k−1_k +γ−d−¹_kε^∗

t⁰.

We would like to work with a regular partition ofV whose number of clusters is a multiple of k. For this purpose we move at mostk−1 of the clusters of U⁰ to U₀⁰ in order to obtain a partition U of V with kr equally sized clusters, where r ∈ N and r⁰₀ ≤ kr ≤ t⁰. By construction, U is an (ε^∗, d, p)-regular partition with exceptional set U₀ of size at most ε^∗n and with a reduced graphR^k_r of minimum degree at least ^k−_k¹ +γ−d−_k¹ε^∗

kr−k.

By the choice ofdand ε^∗ as well as ofr₀⁰ we have δ(R^k_r)≥ ^k−_k¹ +γ−d−¹_kε^∗

kr−k≥ ^k−_k¹+ ^γ₂ kr .

Observe that the bandwidth graph B_r^k on the vertex set [r]×[k] has bandwidth at most 2k < βr₀⁰ and maximum degree less than 3k. Moreover, note that |V(B_r^k)|=kr ≥r⁰₀ ≥n₀ by choice ofr⁰₀. Thus Theorem 1.3, with input γ/2, 3k, and k, states in particular that R^k_r contains a copy of B_r^k. We fix one such copy and let its vertices {(i, j)}i∈[r],j∈[k] label the vertices ofR^k_r. Similarly, for eachi∈[r] andj∈[k], we denote the cluster ofU corresponding to the vertex (i, j) of B^k_r by U_i,j. The partition U ={U_i,j}i∈[r],j∈[k] is equitable and thus in particulark-equitable.

Starting with an empty set, we create a subset Z₁ ⊆V as follows. First we move to Z₁ every vertexv∈V for which

• there exist pairs of indices (i, j),(i⁰, j⁰)∈[r]×[k] with{(i, j),(i⁰, j⁰)} ∈E(R^k_r) such that N_Γ(v, U_i,j), U_i0,j⁰

or N_Γ(v, U_i,j), N_Γ(v, U_i0,j⁰)

is not ε/2, d, p

G-regular, or

• there exists a cluster Ui,j ∈ U with deg_Γ(v, Ui,j)6= (1±ε^∗)p|Ui,j|, or

• deg_Γ(v, U₀)>2ε^∗pn holds.

We also add a minimum number of vertices toZ₁ in order to obtain k-equitability of the sets U_i,j\Z_{1 i}

∈[r],j∈[k]. By Lemmas 2.10 and 2.11, each with inputε/2 andd, and Proposition 3.8 with inputε^∗/(1000k²)< ε^∗ we have

|Z1| ≤4kr²₁Cmax

p⁻², p⁻¹logn ≤ _kr^ε∗₁n , (3.1) where the factorkaccounts for vertices removed to maintain k-equitability.

As a next step, for everyi∈[r] andj∈[k] we collect inW_i,j all vertices fromU_i,j\Z₁that destroyed super-regularity on the copy of K_r^k inB_r^kbefore the removal ofZ₁. More precisely, for each i ∈ [r] and j ∈ [k] let Wi,j be the set of vertices in Ui,j \Z1 that have each less than (d−2ε^∗)p|U_i,j0| neighbours in U_i,j0 for some j⁰ 6=j. Since for each i∈[r] and distinct j, j⁰ ∈[k] the pair (U_i,j, U_i,j0) is (ε^∗, d, p)_G-regular, we have |W_i,j| ≤kε^∗|U_i,j|.

Now letW ⊆V be a set that consists ofU₀\Z₁andS

i∈[r],j∈[k]W_i,jand a minimum number of additional vertices fromV\Z1 to obtaink-equitability of the sets

Ui,j\(Z1∪W) i∈[r],j∈[k]. By construction, we have

|W| ≤ε^∗n+kr·kε^∗_krⁿ ≤2kε^∗n.

Given any vertex w∈ W, we have in particular that w 6∈Z₁ and hence, for each i ∈[r]

andj ∈[k], it holds that

deg_Γ(w, U₀)≤2ε^∗pn and deg_Γ(w, U_i,j)≤(1 +ε^∗)p|U_i,j|.

Now let us consider the edges inE(G) that are incident tow. At most 2ε^∗pnof these go toU0, and, clearly, at most 2dpnsuch edges go to clustersUi,j with deg_G(w, Ui,j)≤2dp|Ui,j|. Since deg_G(w) ≥ ^k−1_k +γ

pnand d≤γ/4, at least ^k−1_k + ^γ₂

pn edges leavingw go to sets U_i,j with deg_G(w, U_i,j) ≥ 2dp|U_i,j|. Since for each cluster U_i,j ∈ U we have |U_i,j| ≤ n/(kr) and deg_G(w, U_i,j) ≤ deg_Γ(w, U_i,j) ≤(1 +ε^∗)p|U_i,j| as w /∈ Z₁, the number of sets U_i,j with i∈[r] andj ∈[k] and deg_G(w, U_i,j)≥2dp|U_i,j|is at least

k−1k + ^γ₂ pn

(1 +ε^∗)p_krⁿ ≥ ^k−1_k +^γ₄ kr.

It follows that there are at leastγr/4 indicesi ∈[r] with deg_G(w, U_i,j) ≥2dp|U_i,j| for each j ∈[k].

To each w ∈W we know assign sequentially an index c(w) ∈[r]×[k], where we choose c(w) = (i, j) as follows. The indexiis chosen minimal in [r] such that we have deg_G(w, U_i,j0)≥ 2dp|U_i,j0| for each j⁰ ∈ [k] and at most 100r⁻¹kε^∗γ⁻¹n vertices w⁰ ∈ W have so far been assignedc(w⁰) = (i, j⁰) for anyj⁰ ∈[k]. We choose j∈[k] minimising the number of vertices w⁰∈W withc(w⁰) = (i, j). Because |W| ≤2kε^∗n, this assignment is always possible.

Next, for eachi∈[r] and j∈[k], we let V_i,j⁰ =Ui,j\(Z1∪Wi,j)∪

w∈W : c(w) = (i, j) . By construction, we have for eachi∈[r] and j∈[k] that

|U_i,j4V_i,j⁰ | ≤ |Z₁|+|W_i,j|+ ¹⁰⁰_r kε^∗γ⁻¹n≤1000k²ε^∗γ⁻¹|U_i,j|. Finally, we letZ₂ consist of all vertices v∈V \Z₁ with

deg_Γ(v, U_i,j4V_i,j⁰ )≥2000k²ε^∗γ⁻¹p|U_i,j|for somei∈[r] and j∈[k],

together with a minimum number of additional vertices ofV \Z₁ to obtaink-equitability of the setsV_i,j⁰ \Z2. For eachi∈[r] and j∈[k] we set

V_i,j =V_i,j⁰ \Z₂ andV₀ =Z₁∪Z₂. We claim thatV :={V_i,j}i∈[r],j∈[k] is the desired partition ofV \V₀.

Note that the setsV_i,j⁰ andV_i,j⁰ ₀ differ in size by at most one for any i∈[r] andj, j⁰ ∈[k], by our construction of the assignmentc. By Proposition 3.8 and thanks to the choice of C^∗ we thus have

|Z₂| ≤r₁+Ckr₁p⁻¹logn≤ _kr^ε∗₁pn . (3.2) This gives

|U_i,j4V_i,j| ≤ |U_i,j4V_i,j⁰ |+|Z₂| ≤2000k²ε^∗γ⁻¹|U_i,j|. (3.3) Given any v ∈ V \ V₀, for each i ∈ [r] and j ∈ [k], we have deg_Γ(v, U_i,j4V_i,j⁰ ) ≤ 2000k²ε^∗γ⁻¹p|U_i,j|becausev /∈Z₂ ⊆V₀. We thus have

deg_Γ(v, U_i,j4V_i,j)≤2000k²ε^∗γ⁻¹p|U_i,j|+|Z₂| ≤3000k²ε^∗γ⁻¹p|U_i,j|. (3.4) Since v 6∈ Z₁ ⊆ V₀ we have deg_Γ(v, U_i,j) = (1±ε^∗)p|U_i,j|, and hence by Equations (3.3) and (3.4)

deg_Γ(v, V_i,j) = 1±10000k²ε^∗γ⁻¹

p|V_i,j|. (3.5)

3.1. The bandwidth theorem in random graphs 55

Adding up (3.1) and (3.2), we get the following desired upper bound on the size of V₀ by choice ofC^∗:

|V₀| ≤4kr²₁Cmax{p⁻², p⁻¹logn}+r₁+Ckr₁p⁻¹logn≤C^∗max{p⁻², p⁻¹logn}, as desired. Furthermore, the partitionV ={V_i,j}i∈[r],j∈[k]is by constructionk-equitable, and the graph R^k_r has minimum degree (k−1)/k+γ/2

kr as desired.

In the remainder of the proof we check that V satisfies Properties (G1)–(G4) as well as the stronger bound |V₀| ≤C^∗p⁻¹logn in case we require (G3’) instead of (G3).

For eachi∈[r] andj ∈[k] we have |U_i,j|= (1±ε^∗)_krⁿ, and so by Equation (3.3) and by our choice ofε^∗ we have

n

4kr ≤(1−ε^∗)(1−2000k²ε₁γ⁻¹)_krⁿ ≤ |V_i,j| ≤(1 +ε^∗)(1 + 2000k²ε^∗γ⁻¹)_krⁿ ≤ ⁴ⁿ_kr, which is Property (G1).

Next, if{(i, j),(i⁰, j⁰)}is an edge ofR^k_r, then (U_i,j, U_i0,j⁰) is (ε^∗, d, p)_G-regular by construc-tion. By (3.3), we have|U_i,j004V_i,j00| ≤2000k²ε^∗γ⁻¹|U_i,j00|forj⁰⁰ ∈ {j, j⁰}and hence we know by Proposition 2.4 that G is (ε, d, p)-regular on (V_i,j, V_i0,j⁰) since ε^∗+ 4p

2000k²ε^∗γ⁻¹ ≤ ε.

Given i ∈ [r] and distinct indices j, j⁰ ∈ [k], let v be a vertex of V_i,j. Observe that since v ∈ V_i,j, either we have v ∈ U_i,j \W or v ∈ W. In the first case, since v 6∈ W, we have deg_G(v, U_i,j0) ≥ (d−2ε^∗)p|U_i,j|. In the other case, it holds that c(v) = (i, j) and hence deg_G(v, U_i,j0)≥dp|U_i,j0|. By (3.3) and (3.4) we have

deg_G(v, V_i,j0)≥(d−2ε^∗)p|U_i,j| −3000k²ε^∗γ⁻¹p|U_i,j| ≥(d−ε)p|V_i,j0|, giving (G2).

Let {(i, j),(i⁰, j⁰)} ∈ E(R^k_r). Then for any v ∈ V \V₀, since v 6∈ Z₁, we know that the pairs N_Γ(v, U_i,j), U_i0,j⁰

and N_Γ(v, U_i,j), N_Γ(v, U_i0,j⁰)

are ε/2, d, p

G-regular. By (3.4) and since v /∈Z₁ we have for (i⁰⁰, j⁰⁰)∈ {(i, j),(i⁰, j⁰)}that

|N_Γ(v, U_i00,j⁰⁰)4N_Γ(v, V_i00,j⁰⁰)| ≤3000k²ε^∗γ⁻¹p|U_i00,j⁰⁰| ≤6000k²ε^∗γ⁻¹|N_Γ(v, U_i00,j⁰⁰)|. Using this fact and Equation (3.3) we know by Proposition 2.4 that both NΓ(v, Vi,j), V_i0,j⁰

and NΓ(v, Vi,j), NΓ(v, V_i0,j⁰)

are ε, d, p

G-regular since ε/2 + 4p

6000k²ε^∗γ⁻¹ ≤ ε. This shows Property (G3).

Finally, (G4) follows directly from (3.5) and our choice ofε^∗.

If we alter the definition of Z₁ by removing the condition on N_Γ(v, U_i,j), N_Γ(v, U_i0,j⁰) , then we do not need to use Lemma 2.11 and the bound in (3.1) therefore improves to|Z₁| ≤ 3kr₁²Cp⁻¹logn. Thus, if we only require (G3’), we obtain|V0| ≤C^∗p⁻¹lognas claimed.

Im Dokument Existence and Enumeration of Spanning Structures in Sparse Graphs and Hypergraphs (Seite 62-65)