Sparse regular partitions

An essential concept of the regularity method is the notion of regular pairs. LetG= (V, E) be a graph, and letε, d > 0 and p ∈(0,1] be reals. Moreover, letX, Y ⊆V be two disjoint nonempty sets. Thep-density of the pair (X, Y) is defined as

d_G,p(X, Y) := e_G(X, Y) p|X||Y| .

We give two definitions of regularity. The first one requires a lower bound on the p-density of all subpairs of a certain size.

Definition 2.1 ((Super-)regular pairs). The pair (X, Y) is called (ε, d, p)_G-regular if for every X⁰⊆X and Y⁰ ⊆Y with |X⁰| ≥ε|X|and |Y⁰| ≥ε|Y| we have

d_G,p(X⁰, Y⁰)≥d−ε.

If additionally we have |N_G(x, Y)| ≥ (d−ε)p|Y| and |N_G(y, X)| ≥ (d−ε)p|X| for every x∈X andy∈Y, then the pair (X, Y) is called (ε, d, p)_G-super-regular.

The second definition additionally imposes an upper bound on the p-density of those subpairs.

Definition 2.2 ((Super-)fully-regular). The pair (X, Y) is called (ε, d, p)_G-fully-regular if there existsd⁰ ≥d such that for every X⁰ ⊆X and Y⁰ ⊆Y with |X⁰| ≥ε|X| and |Y⁰| ≥ε|Y| we have

d_G,p(X⁰, Y⁰) =d⁰±ε.

If additionally we have |N_G(x, Y)| ≥ (d−ε)p|Y| and |N_G(y, X)| ≥ (d−ε)p|X| for every x∈X andy∈Y, then the pair (X, Y) is called (ε, d, p)G-super-fully-regular.

Most of the time we will use the first version of regularity, which is sometimes called lower-regularity. This is the version we have to use when dealing with subgraphs of random graphs. However, the sparse regularity lemma gives us fully-regular pairs, and we have to work with this stronger concept when dealing with subgraphs of bijumbled graphs. Note that an (ε, d, p)-fully-regular pair is in particular (ε, d, p)-regular.

Whenever there is no risk of confusion, we might omit the subscriptGin (ε, d, p)_G-(super-) (fully-)regular, which indicates with respect to which graph a pair is (super-)(fully-)regular.

A direct consequence of the definition of (ε, d, p)-regular pairs is the following proposition about the sizes of neighbourhoods in regular pairs.

Proposition 2.3. Let(X, Y)be(ε, d, p)-regular. Then there are less thanε|X|verticesx∈X with|N(x, Y)|<(d−ε)p|Y|.

The next proposition asserts that small alterations of the vertex sets of an (ε, d, p)-regular pair do not destroy regularity.

Proposition 2.4. Let (X, Y) be an (ε, d, p)-regular pair in a graph G and let Xˆ and Yˆ be two subsets of V(G) such that |X4Xˆ| ≤ µ|X| and |Y4Yˆ| ≤ ν|Y| for some 0 ≤ µ, ν ≤ 1.

Then( ˆX,Yˆ) is (ˆε, d, p)-regular, where εˆ:=ε+ 2√µ+ 2√

ν. Furthermore, if for any disjoint subsetsA, A⁰ ⊆V(G) with|A| ≥µ|X|and|A⁰| ≥ν|Y|we have e(A, A⁰)≤(1 +µ+ν)p|A||A⁰|, and (X, Y) is (ε, d, p)-fully-regular, then ( ˆX,Yˆ) is(ˆε, d, p)-fully-regular.

Proof. Let A ⊆ Xˆ and B ⊆ Yˆ such that |A| ≥ εˆ|Xˆ| and |B| ≥ εˆ|Yˆ| be given. Define A⁰ :=A∩X and B⁰ :=B∩Y and note that

|A⁰| ≥ |A| −µ|X| ≥εˆ|Xˆ| −µ|X| ≥ε(1ˆ −µ)|X| −µ|X| ≥ εˆ−2√µ

|X| ≥ε|X| by the definition of ˆε. Analogously, one can show that |B⁰| ≥ ε|Y|. Since (X, Y) is an (ε, d, p)-regular pair, we know that d_p(A⁰, B⁰)≥d−ε. Furthermore, we have

|A⁰| ≥ |A| −µ|X| ≥ |A| −µ|A| ˆ

ε ≥ 1−√ µ

|A| and by an analogous calculation we get|B⁰| ≥ 1−√

ν

|B|. For the number of edges between A and B we get

e(A, B)≥e(A⁰, B⁰)≥(d−ε)p|A⁰||B⁰| ≥(d−ε)p 1−√µ

1−√ ν

|A||B|

≥ d−ε−2√µ−2√ ν

p|A||B| ≥(d−ε)pˆ |A||B|. Therefore we have

d_p(A, B)≥d−ε.ˆ

2.2. The regularity method 31

Now suppose that (X, Y) is (ε, d, p)-fully-regular. Let d⁰ be such that d_p(A⁰, B⁰) =d⁰±ε for anyA⁰ ⊆X and B⁰ ⊆Y with |A⁰| ≥ε|X| and |B⁰| ≥ε|Y|. Let A⊆Xˆ and B ⊆Yˆ with

|A| ≥εˆ|Xˆ|and |B| ≥εˆ|Yˆ|be given. As above, we obtain e(A, B)≥(d⁰−ε)pˆ |A||B|. We also have

e(A, B)≤e(A⁰, B⁰) +e(A⁰, B\B⁰) +e(A\A⁰, B)

≤(d⁰+ε)p|A⁰||B⁰|+ (1 +µ+ν)p|A⁰|ν|B|+ (1 +µ+ν)pµ|A||B|

≤(d⁰+ ˆε)|A||B|,

so that ( ˆX,Yˆ) is (ˆε, d, p)-fully-regular, as desired.

A partition V = {V_i}i∈{0,...,r} of the vertex set of G is called an (ε, p)_G-(fully-)regular partition ofV(G) if|V₀| ≤ε|V(G)|and (V_i, V_i0) forms an (ε, p)-(fully-)regular pair inGfor all but at mostε ^r₂

pairs {i, i⁰} ∈ ^[r]₂

. The partitionV is called (ε, d, p)-(super-)(fully-)regular on a graph R = ([r], F) if |V₀| ≤ ε|V(G)| and (V_i, V_i0) is (ε, d, p)-(super-)(fully-)regular for every{i, i⁰} ∈F. The graphRis referred to as thereduced graph ofV, the partition classesV_i withi∈[r] asclusters, andV₀ as theexceptional set. We callV anequipartition if|V_i|=|V_i0| for every i, i⁰ ∈[r].

Analogously to Szemer´edi’s regularity lemma for dense graphs, the sparse regularity lemma, proved by Kohayakawa and R¨odl [105, 106], asserts the existence of an (ε, p)-fully-regular partition of constant size of any sparse graph. We need two versions of this lemma in our proofs. The first one allows an initial partition with parts of different sizes to be equitably refined.

Before stating this lemma, we require one more definition. Given a partition{V_i}i∈[s] of the vertex set of a graphG, we say that a partition{V_i,j}i∈[s],j∈[t]is anequitable(ε, p)-regular refinement of{V_i}i∈[s]if|V_i,j|=|V_i,j0| ±1 for eachi∈[s] andj, j⁰ ∈[t], and there are at most εs²t² pairs (Vi,j, V_i0,j⁰) that are not (ε, p)-regular.

Lemma 2.5. For each ε > 0 and s ≥ 1 there exists t₁ ≥ 1 such that the following holds for every 0 < p < 1. Given any graph G, suppose {Vi}i∈[s] is a partition of V(G). If e(V_i) ≤ 3p|V_i|² for each i ∈ [s], and e(V_i, V_i0) ≤ 2p|V_i||V_i0| for all distinct i, i⁰ ∈ [s], then there exist sets V_i,0 ⊆ V_i for each i ∈ [s] with |V_i,0| < ε|V_i|, and an equitable (ε, p)-regular refinement {V_i,j}i∈[s],j∈[t] of {V_i\V_i,0}i∈[s] for some t≤t₁.

The proof of Lemma 2.5 follows the proof of a sparse regularity lemma by Scott [147].

Proof of Lemma 2.5. Given ε > 0 and s ≥ 1, set L = 100s²ε⁻¹. Let n₁ = 1, and for each j ≥1 letnj+1= 10000ε⁻¹nj2^snj. Finally, sett1 =n_1000ε−5(L²+16Ls²)+1.

For all i, i⁰ ∈ [s], we define the energy E(P, P⁰) of a pair of disjoint subsets P ⊆ V_i and P⁰ ⊆V_i0 to be

E(P, P⁰) = |P||P⁰|min

d_G,p(P, P⁰)²,2L·d_G,p(P, P⁰)−L²

|V_i||V_i0| .

Note that this quantity is convex in d_G,p(P, P⁰). Given a partition P refining {V_i}i∈[s], we define the energyE(P) of P to be

E(P) := X

{P,P⁰}⊆P

E(P, P⁰).

We now construct a succession of partitionsPj+1 for eachj≥1, each refiningP1:={V_i}i∈[s], and claim that the following holds for everyj≥2:

(R1) Pj partitions every set V_i ∈ {V_i0}i⁰∈[s] into between n_j and 1 + 10⁻⁴ε

n_j sets such that the largest nj sets are equally sized.

(R2) E(P2)≥10⁻³ε⁵j.

We stop ifPj is (ε/2, p)-regular. If not, then we apply the following procedure.

For each pair of Pj that is not (ε/2, p)-regular, we take a witness of its irregularity, consisting of a subset of each side of the pair. We letP_j⁰ be the union of the Venn diagrams of all witness sets in each part ofPj. SincePj is not (ε/2, p)-regular, there are at least ¹₂εs²n²_j pairs that are not (ε/2, p)-regular. By choice of Land by (R1), at least ¹₄εs²n²_j of these pairs have density at mostL/2. By the defect Cauchy-Schwarz inequality, just from refining these pairs we conclude thatE(Pj⁰)≥ E(Pj) + 10⁻³ε⁵ (cf. [147]). Note that, by convexity ofE(P, P⁰) ind_G,p(P, P), refining the other pairs does not affect E(P_j⁰) negatively.

We now let Pj+1 be obtained by splitting each set of Pj⁰ within each V_i into sets of size

1000−ε

1000nj+1|V_i| plus at most one smaller set. By Jensen’s inequality, we have E(Pj+1) ≥ E(Pj⁰) (cf. [147]), giving (R2). Since Pj⁰ partitions each V_i into at most n_j2^snj = 10⁻⁴εn_j+1, the total number of smaller sets is at most 10⁻⁴εn_j+1. This gives (R1).

Observe that for any partitionP refiningP1, we haveE(P)≤L²+ 16Ls². It follows that this procedure must terminate withj ≤1000ε⁻⁵(L²+ 16Ls²) + 1. The final partition Pj is thus (ε/2, p)-regular. For each i∈ [s], let Vi,0 consist of the union of all but the largest nj

parts ofPj. LetP be the partition ofS

i∈[s]V_i\V_i,0 given byPj. This is the desired equitable (ε, p)-regular refinement of {V_i\V_i,0}i∈[s].

The second variant of the sparse regularity lemma that we need is the following minimum degree version.

Lemma 2.6 (Minimum degree version of the sparse regularity lemma). For each ε > 0 and r₀ ≥ 1 there exists r₁ ≥ 1 with the following property. For any d ∈ [0,1], any positive reals α andp, and any n-vertex graph Gwith minimum degree αpnsuch that for any disjoint X, Y ⊆V(G)with|X|,|Y| ≥ ^εn_r1 we havee(X, Y)≤ 1+₁₀₀₀¹ ε²

p|X||Y|, there is an(ε, d, p)_G -fully-regular equipartition of V(G) with reduced graph R such that δ(R)≥(α−d−ε)|V(R)| andr0≤ |V(R)| ≤r1.

Using Lemma 2.5 we now give a proof of Lemma 2.6, which follows [39] (c.f. [105]).

Proof of Lemma 2.6. Givenε >0 andr₀≥1, without loss of generality we assumeε≤1/10.

Lett₁ be returned by Lemma 2.5 for input ε²/(1000s) ands= 100r₀/ε. Setr₁ =st₁. Givenα >0 andp >0, letGbe ann-vertex graph with minimum degreeαpn. Let{V_i}i∈[s]

be an arbitrary partition of V(G) into sets of as equal size as possible. By assumption, we have e(V_i, V_i0) ≤ 2p|V_i||V_i0| for distinct indices i, i⁰ ∈ [s]. Furthermore, if V_i is a part with e(V_i)≥3p|V_i|², then for a maximum cut (A, A⁰) ofV_iwe havee(A, A⁰)≥3p|V_i|²/2. Enlarging the smaller of the sets A and A⁰ if necessary, we have a pair of subsets ofV(G) both of size at most |V_i| between which there are at least 3p|V_i|²/2 edges, contradicting the assumption of Lemma 2.6. Thus G satisfies the conditions of Lemma 2.5 with input ε²/(1000s) and s.

Applying that lemma, we obtain a collection{V_i,0}i∈[s]of sets, and an (ε, p)-regular partition

2.2. The regularity method 33

P ofS

i∈[s]V_i\V_i,0, which partitions eachV_i\V₀ intot≤t₁ sets. Note thatr₀ ≤s≤ |P| ≤r₁ by construction.

Now letV₀⁰ be the union of theV_i,0 for i∈[s], of each set W ∈ P that lies in more than εst/4 pairs which are not (ε/1000, p)-regular, and at most two vertices from each set W ∈ P in order that the partition ofV(G)\V₀⁰ induced by P is an equipartition. Because the total number of pairs that are not (ε/1000, p)-regular is at mostε²/(1000sr₀²t²), the number of such sets in any givenVi is at mostεt/100, so|V_i,0⁰ | has size at mostε|Vi|/50, and the number of parts ofP inV_i\V_i,0⁰ is larger thant/2. Thus the partition P⁰ ofV(G)\V₀⁰ induced by P is an (ε, p)-regular equipartition ofV(G)\V₀⁰, and we have |V₀⁰| ≤εn.

We claim that the partitionP⁰ has all the properties we require. It remains to verify that for eachd∈[0,1], thed-reduced graph ofP⁰has minimum degree at least (α−d−ε)t⁰. Suppose thatP is a part of P⁰. Now we havee(P) ≤3p|P|², since otherwise, as before, a maximum cut (A, A⁰) of P has at least 3p|P|²/2 < εp|P|n/20 edges, yielding a contradiction to the assumption on the maximum density of pairs of G. By construction, P lies in at most εt⁰/2 pairs that are not (ε, p)-regular, and these contain at most (1+ε/10)p|P| εt⁰|P|/2

< ³₄εp|P|n edges ofG. We conclude that at least αp|P|n− ⁷₈εp|P|n edges of G leaving P lie in (ε, p)-regular pairs ofP⁰. Of these, at most dp|P|ncan lie in pairs of density less than p, so that the remaining at least α−d−⁷₈ε

p|P|nedges lie in (ε, d, p)-regular pairs. If so many edges were in less than (α−d−ε)t⁰ pairs leaving P, this would contradict our assumption on the maximum density ofG, so that we conclude thatP lies in at least (α−d−ε)t⁰ pairs that are (ε, d, p)-regular, as desired.

Im Dokument Existence and Enumeration of Spanning Structures in Sparse Graphs and Hypergraphs (Seite 39-43)