Balancing lemma

The statement of Lemma 3.7 gives us a partition ofV(G) with parts

Vi,j i,j, and a collection properties as the original partition, while not substantially changing common neighbourhoods of vertices.

There are two steps to our proof. In a first step, we correct global imbalance, that is, we find a partition Ve, which maintains all the desired properties and which has the property that P

vertices fromV_1,j∗ to some clusterV_i0,j⁰, maintaining the desired properties, and repeat this procedure until no global imbalance remains.

In a second step, we correctlocal imbalance, that is, for eachi= 1, . . . , r−1 sequentially, and for each j ∈ [k], we move vertices between Ve_i,j and Ve_i+1,j, maintaining the desired properties, in order to obtain a partition V⁰ such that |V_i,j⁰ | = n_i,j for each i ∈ [r−1] and j∈[k]. Observe that, since Ve is globally balanced, once we know that|V_i,j⁰ |=n_i,j holds for eachi∈[r−1] andj ∈[k], we are guaranteed that|V_r,j⁰ |=n_r,j holds as well for each j∈[k].

The proof of the lemma then comes down to showing that we can move vertices and maintain the desired properties. Because we start with a partition in which|V_i,j|is very close to n_i,j for each i ∈ [r] and j ∈ [k], the total number of vertices we move in any step is at

most the sum of the differences, which is much smaller than anyn_i,j. The following lemma shows that we can move any small (compared to allni,j) number of vertices from one part to another and maintain the desired properties.

Lemma 3.12. For all integers k, r₁,∆ ≥ 1, and reals d > 0 and 0 < ε < 1/2k as well as 0< ξ <1/(100kr³₁), there exist C^∗ >0 such that the following holds for sufficiently large n.

Let Γ be a graph on vertex set [n], and let G be a not necessarily spanning subgraph. Let X, Z₁, . . . , Z_k−1 ⊆V(G)be pairwise disjoint subsets, each of size at leastn/(16kr₁), such that (X, Zi) is(ε, d, p)_G-regular for each i∈[k−1]. Then for each 1≤m≤2r²₁ξn, there exists a setS of m vertices of X with the following properties.

(SM1) For each v∈S we have deg_G(v, Zi)≥(d−ε)p|Zi|for each i∈[k−1], and (SM2) for each 1≤s≤∆and every collection of vertices v₁, . . . , v_s ∈[n] we have deg_Γ(v₁, . . . , v_s;S)≤100kr₁³ξdeg_Γ(v₁, . . . , v_s;X) + ₁₀₀¹ C^∗logn .

Proof. Givenk,r₁, ∆,d,ξ andε. LetC be returned by Lemma 2.19 for input ξ and ∆. We setC^∗= 100C. Given Γ,Gand X,Y,Z₁, . . . , Z_k−1, let X⁰ be the set of verticesv∈X such that deg_G(v;Zi)≥(d−ε)p|Zi|for eachi∈[k−1]. Because each pair (X, Zi) fori∈[k−1]

is (ε, d, p)_G-regular, we have |X⁰| ≥ |X| −kε|X| ≥ |X|/2.

We now apply Lemma 2.19, with input ξ, ∆, W = X⁰ and the sets T_i consisting of the sets NΓ(v1, . . . , vs;X⁰) for each 1 ≤ s ≤ ∆ and v1, . . . , vs ∈ [n], to choose a set S of size m≤2r₁²ξn≤ |X⁰|inX⁰. We then have

deg_Γ(v₁, . . . , v_s;S)≤_2r2 1ξn

|X⁰| +ξ

deg_Γ(v₁, . . . , v_s;X⁰) +Clogn

≤100kr₁³ξdeg_Γ(v1, . . . , vs;X) +₁₀₀¹ C^∗logn ,

where the final inequality is by choice ofC^∗, and since |X⁰| ≥ |X|/2≥n/(32kr₁). Thus the setS satisfies (SM2), and since S⊆X⁰ we have (SM1).

We now turn to the proof of the balancing lemma.

Proof of Lemma 3.7. Given integers k, r₁,∆ ≥ 1 as well as reals γ, d > 0 and 0 < ε <

min{d,1/(2k)}, we set

ξ= 10⁻¹⁵ε⁴d/(k³r₁⁵).

LetC₁^∗ be returned by Lemma 3.12 with inputk,r₁, ∆,d,ε/4, andξ, and letC₂^∗ be returned by Lemma 3.12 with inputk,r1, ∆,d, 3ε/4, andξ. We setC^∗ = max{C₁^∗, C₂^∗}.

Now suppose that p ≥ C^∗(logn/n)^1/2, that 10γ⁻¹ ≤ r ≤ r₁, and that graphs Γ and G, a partition V of V = V(G), and graphs R^k_r, B_r^k and K_r^k on [r]×[k] as in the statement of Lemma 3.7 are given. We divide the proof into two stages.

First stage (global imbalance):

The goal of the first stage is to move vertices between clusters ofV such that the partition Ve = {Ve_i,j}i∈[r],j∈[k] that we obtain satisfies P

i∈[r]|Ve_i,j| = P

i∈[r]n_i,j for every j ∈ [k], such thatV is (ε/2, d, p)_G-regular on R^k_r and (ε/2, d, p)_G-super-regular on K_r^k, and such that the sizes of the clusters and the common Γ-neighbourhoods of any at most ∆ vertices restricted to any cluster has not changed much.

3.1. The bandwidth theorem in random graphs 67

We use the following algorithm to create a globally balanced partition Ve, i.e. for which P

i∈[r]|Vei,j|=P

i∈[r]ni,j holds. In Claim 3.13 we show that the properties mentioned above hold for every partition that is the outcome of the algorithm if we select the set S in the fourth step of the ‘While loop’ cleverly. A cluster that has not been changed by Algorithm 1 before is referred to as ’unchanged’. Figure 3.3 for an illustration of Algorithm 1.

V_1,1

Figure 3.3: One iteration step in Algorithm 1, where vertices ofV_1,j∗ are moved to V_i0,j⁰. We claim that the algorithm completes successfully. In other words, we show that each of the choices is possible and that Lemma 3.12 is always applicable. In each ‘While loop’, sinceP

i∈[r],j∈[k](|V_i,j| −n_i,j) = 0 and since the While condition is satisfied, the indexj^∗∈[k]

maximisingP

i∈[r](|V_i,j∗| −n_i,j) satisfiesP

i∈[r](|V_i,j∗| −n_i,j∗)>0.

Observe that the ‘While loop’ is run at mostk times since at the end of the ‘While loop’

we have achieved P

i∈[r](|Vi,j| −ni,j) = 0 for an index j ∈ [k] and do not select this index in future iterations as j^∗ or j⁰. It follows that the number of clusters flagged as ‘changed’

never exceeds 2k. Every vertex (1, j^∗)∈ {1} ×[k] has degree at least (k−1 +γk/2)r inR^k_r.

-regular. Thus it is possible to choosei⁰ in the second step of the ‘While loop’. It is possible to choose j⁰ ∈[k] in the third step of the ‘While loop’ sinceP

i∈[r],j∈[k](|Vi,j| −ni,j) = 0 and P

i∈[r](|V_i,j∗| −n_i,j∗)>0.

Finally, we need to show that Lemma 3.12 is always applicable with the given param-eters. In each application, the sets denoted by X, Z₁, . . . , Z_k−1 are parts of the partition V. This means that they have not been changed by the algorithm yet. It follows that each set has size at least n/(8kr) > n/(16kr₁). Since V is (ε/4, d, p)_G-regular on B^k_r, the pairs (X, Z₁), . . . ,(X, Z_k−1) are (ε/4, d, p)_G-regular as required. Finally, by choice of j^∗ ∈ [k] we see that the sizes of the setsS that we select in each step are decreasing. Hence it is enough to show that in the first step we have|S| ≤rξn, which follows from (B1). Thus Lemma 3.12 is applicable in each step, and we conclude that the algorithm indeed completes. We denote the resulting vertex partition byVe ={Vei,j}i∈[r],j∈[k].

Claim 3.13. The following properties hold forVe: (P1) P

i∈[r]|Ve_i,j|=P

i∈[r]n_i,j,

(P2) for each i∈[r] andj ∈[k]we have |Ve_i,j| −n_i,j≤2rξn, (P3) Ve is ₂^ε, d, p

G-regular on R^k_r and ^ε₂, d, p

G-super-regular on K_r^k, (P4) for each i∈[r], j∈[k]and 1≤s≤∆and v₁, . . . , v_s∈[n] we have

|N_Γ(v1, . . . , vs;Vei,j)4N_Γ(v1, . . . , vs;Vi,j)| ≤100kr³₁ξdeg_Γ v1, . . . , vs;V(G)

+₁₀₀¹ C^∗logn . Proof. Property (P1) holds by construction of Algorithm 1.

Observe that vertices were removed from or added to each V_i,j to form Ve_i,j at most once in the running of Algorithm 1, and the number of vertices added or removed was at most rξn. Since|V_i,j|satisfies (B1), we conclude that (P2) holds. Furthermore, the vertices added to or removed fromV_i,j satisfy (SM2) and therefore (P4) holds.

Since |Vi,j| ≥ n/(8kr) for each i ∈ [r] and j ∈ [k], we can apply Proposition 2.4 with µ = ν = 8kr²ξ to each edge of R^k_r, concluding that Ve is (ε/2, d, p)_G-regular on R^k_r since ε/4 + 4p

8kr²ξ < ε/2. Now for any i∈[r] and j ∈[k], consider v∈Ve_i,j. If v6∈V_i,j, then we applied Lemma 3.12 to selectv, and at that time noV_i,j0 withj⁰ ∈[k] was flagged as changed by Algorithm 1. Thus by (SM1) we have

deg_G(v;Ve_i,j0) = deg_G(v;V_i,j0)≥ d−^ε₄

p|V_i,j0|= d−^ε₄ p|Ve_i,j0|

for each j⁰ ∈ [k]\ {j} since V_i,j is then flagged as changed and thus V_i,j0 = Ve_i,j0 for each j⁰ ∈ [k]\ {j}. If on the other hand v ∈ V_i,j, then by (B2) we started with deg_G(v;V_i,j0) ≥ (d−ε/4)p|V_i,j0|. By (SM2) and (B4), we have

deg_G(v;Ve_i,j0)≥ d−^ε₄

p|V_i,j0| − _1000kr^ε2 ₁ 1 +₄^ε

p|V_i,j0| −₁₀₀¹ C^∗logn≥ d−^ε₂

p|Ve_i,j0|, where the final inequality follows since|Ve_i,j0| ≤ |V_i,j0|+rξn≤(1+εd/100)|V_i,j0|and sincenwas assumed to be sufficiently large. Therefore, Ve is (ε/2, d, p)-super-regular onK_r^k, giving (P3).

3.1. The bandwidth theorem in random graphs 69

Second stage (local imbalance):

The first stage resulted in a partition Ve with Properties (P1)–(P4) of Claim 3.13. In particular,Ve is globally balanced, i.e.P

i∈[r]|Ve_i,j|=P

i∈[r]n_i,j for everyj∈[k]. The goal of the second stage is to obtain a balanced partitionV with the desired properties of the lemma.

We use the following algorithm to correct the local imbalances inVe. Algorithm 2:Local balancing

foreachi= 1, . . . , r−1 do foreachj = 1, . . . , k do if |Ve_i,j|> n_i,j then

Select S⊆Ve_i,j with|S|=|Ve_i,j| −n_i,j ;

Update Vei,j :=Vei,j\S and Vei+1,j:=Vei+1,j∪S ; end

else

Select S⊆Ve_i+1,j with|S|=n_i,j− |Ve_i,j|; Update Ve_i+1,j:=Ve_i+1,j\S and Ve_i,j :=Ve_i,j∪S ; end

end end

Again, in each step when we select S we make use of Lemma 3.12 to do so. If we select S fromVe_i,j, then we use the input k,r₁,dand 3ε/4 withX =Ve_i,j and the sets Z₁, . . . , Z_k−1 being{Ve_i+1,j0}j⁰∈[r]\{j}. If on the other hand we selectSfromVe_i+1,j, then we use the inputk, r₁,dand 3ε/4, withX =Ve_i+1,j and the setsZ₁, . . . , Z_k−1 being{Ve_i,j0}j⁰∈[r]\{j}. In Figure 3.4 we sketch one iteration step of Algorithm 2.

V˜i,1

V˜i,j

V˜i,k

V˜i+1,1

V˜i−1,1

V˜_i+1,j V˜i−1,j

V˜i+1,k

V˜i−1,k

S

if|V˜i,j| ≤ni,j if|V˜i,j|> ni,j

Figure 3.4: One iteration step in Algorithm 2, where vertices ofVi,jare either moved toVi−1,j

or to V_i+1,j.

We claim that Lemma 3.12 is always applicable. To see that this is true, observe first that the number of vertices that we move between any Ve_i,j and Ve_i+1,j in a given step is, thanks to (P2), bounded by 2r²ξn. We change any given Ve_i,j at most twice in the running of the algorithm, so that in total at most 4r²ξn vertices are changed. In particular, we maintain

|Ve_i,j| ≥n/(16kr₁) throughout. By Proposition 2.4, with inputµ=ν = 4r²ξn/ n/(16kr₁)

<

100r₁³kξ, and by (P3) we maintain the property that any pair inR^k_r, and in particular any pair inB^k_r, is (3ε/4, d, p)-regular throughout. This shows that Lemma 3.12 is always applicable, and therefore the algorithm completes and returns a partitionV⁰ ={V_i,j⁰ }i∈[r],j∈[k].

We claim that V⁰ is the desired partition. To this end, we need to verify that Proper-ties (B1’)–(B5’) hold.

As for each j ∈ [k] we have P

i∈[r]|V_i,j⁰ | = P

i∈[r]|Vei,j| = P

i∈[r]ni,j, and as |V_i,j⁰ | = ni,j

for each i ∈ [r −1] and j ∈ [k], we conclude that |V_i,j⁰ | = n_i,j for all i ∈ [r] and j ∈ [k], giving (B1’).

For the first part of (B3’), we have justified that we maintain (3ε/4, d, p)_G-regularity on R^k_r throughout the algorithm. For the second part, we need to show that for each i ∈ [r]

and distinct indices j, j⁰ ∈ [k], and each v ∈ V_i,j⁰ , we have deg_G(v;V_i,j⁰ ₀) ≥ (d−ε)p|V_i,j⁰ ₀|. If v∈Ve_i,j, then by (P3) we have deg_G(v;Ve_i,j0)≥(d−ε/2)p|Ve_i,j0|. We changeVe_i,j0 at most twice to obtainV_i,j⁰ ₀, both times by adding or removing vertices satisfying (SM2). As in the proof of Claim (P4) above, using (B4) and (P4) we obtain deg_G(v;Ve_i,j0)≥(d−ε)p|V_i,j⁰ |as desired. If v 6∈Ve_i,j, then it was added to the setVe_i,j by Algorithm 2, andVe_i,j0 was changed at most twice thereafter. Again, using (SM1), (SM2), (B4), and (P4) we obtain deg_G(v;Ve_i,j0)≥(d−ε)p|V_i,j⁰ | as desired.

In Algorithm 1 at mostrξnvertices were removed from or added toVi,j. Hence, we have

|V_i,j4Ve_i,j| ≤rξn. In Algorithm 2 at most 4r²ξnare removed from or added toVe_i,j. By choice of ξ this implies|V_i,j4V_i,j⁰ ₀| ≤5r²ξn≤10⁻¹⁰ε⁴k⁻²r⁻²₁ n, which is Property (B2’).

To see that Property (B4’) holds, observe thatthat for any given i ∈ [r] and j ∈ [k] we changeVei,j at most twice in the running of Algorithm 2, both times either adding or removing a set satisfying (SM2). Hence by (B4), (P4), (SM2), and by choice of ξ we have

NΓ(v;Vi,j)∆NΓ(v;V_i,j⁰ )≤ _1000kr^ε2 ₁deg_Γ v;V(G)

+₁₀¹C^∗logn≤ ₁₀₀^ε2 deg_Γ(v;Vi,j) where the final inequality follows by choice of p and of n sufficiently large. Using (B3), we can apply Proposition 2.4, with µ =ν = ε²/100, to deduce (B4’). By a similar calculation Property (B5’) holds.

Finally, suppose that for any two disjoint vertex sets A, A⁰ ⊆ V(Γ) with |A|,|A⁰| ≥ ε²ξpn/(50000kr1) we have eΓ(A, A⁰) ≤(1 +ε²ξ/100)p|A||A⁰|. In each application of Propo-sition 2.4 we haveµ, ν ≥ε²ξ/200, and if we have ‘fully-regular’ in place of ‘regular’ in (B2) and (B3), we always apply Proposition 2.4 to a fully-regular pair with sets of size at least εpn/(1000kr1), so it returns fully-regular pairs for (B3’) and (B4’), as desired.

Im Dokument Existence and Enumeration of Spanning Structures in Sparse Graphs and Hypergraphs (Seite 75-80)