LECTURE 4. REGULARITY LEMMA II
1. Stability : Approximate to exact
By the Erdős-Stone theorem, we know that ex(n, C2k+1 ) = ( 14 − o(1))n2
for all k ≥ 1. We will prove the following strengthening of this result using
the regularity lemma.
Theorem
1. For all k ≥ 1, if n is sufficiently large then ex(n, C2k+1 ) =
j 2k
n
4 .
For a partition Π = V0 ∪ V1 ∪ · · · ∪ Vk of a graph G, we say that an edge
e of G is relevant to the (ε, δ)-reduced graph R(Π) if the two endpoints of
e intersect distinct pairs (Vi , Vj ) that form an ε-regular pair of density at
least δ.
Lemma 2. If Π = V0 ∪V1 ∪· · ·∪Vk is an ε-regular partition of G with k ≥ 1ε ,
then all but at most (3ε + 2δ )n2 edges are relevant to the (ε, δ)-reduced graph
R(Π).
Proof. The following are the non-relevant edges:
(i)
(ii)
(iii)
(iv)
edges incident to V0 (at most εn2 edges),
edges inside Vi for all i (at most k · ( nk )2 ≤ εn2 edges),
edges between irregular pairs (at most εk 2 · ( nk )2 ≤ εn2 edges).
edges between regular pairs of density at most δ (at most δ( nk )2 k2 <
δ 2
2 n edges).
In order to prove the theorem above, we first prove the following stability
theorem.
Theorem 3. For every k ≥ 1 and ε, there exist δ and n0 such that if a
graph G on n vertices is C2k+1 -free and has at least ( 14 − δ)n2 edges, then
one can remove at most εn2 edges to make it bipartite.
Proof. Recall that we previously proved the k = 1 case of this theorem.
Suppose that k ≥ 2. Given ε, let δ be small enough so that every trianglefree graph of at least ( 14 − 2δ)n2 edges can be made bipartite by removing
at most 2ε n2 edges.
Let G be a C2k+1 -free graph on n vertices with at least ( 41 − δ)n2 edges.
Let ε0 be sufficiently small depending on δ and ε. Take an ε0 -regular partition
Π = V0 ∪ V1 ∪ · · · ∪ Vk of G where ε10 ≤ k ≤ T . Note that the (ε0 , 3ε )-reduced
graph R(Π) is triangle-free as otherwise we can find a C2k+1 in G.
1
LECTURE 4. REGULARITY LEMMA II
2
By Lemma 2, there are at most (3ε0 + 3ε )n2 ≤ 2ε n2 edges not relevant to
R(Π). Thus R(Π) is a k-vertex graph with at least
1
ε 2
1
2
−δ n − n ≥
− 2δ n2
4
2
4
edges. Hence R(Π) can be made bipartite by removing at most 2ε k 2 edges.
Therefore by removing all the non-relevant edges and at most ( nk )2 · 2ε k 2 =
ε 2
2 n more edges, we can make G bipartite. Thus G can be made bipartite
by removing at most εn2 edges.
We now prove Theorem 1.
Proof
j 2 k of Theorem 1. The complete bipartite graph shows that ex(n, C2k+1 ) ≥
n
4 . Thus it suffices to prove the other direction of inequality for sufficiently large n. We already proved the theorem for k = 1 and hence we may
assume that k ≥ 2.
For convenience, assume that n is even, and suppose that G is an n-vertex
2
C2k+1 -free graph with at least n4 + m edges for some m > 0. Let A ∪ B
be a partition of V (G) with the maximum number of crossing edges and
let H be the bipartite graph induced on the partition A ∪ B. Note that
eH (A, B) ≥ ( 41 − ε)n2 . If |A| = ( 12 + α)n and |B| = ( 12 − α)n, then
1
1
2
(1)
|A| · |B| =
− α ≥ eH (A, B) >
− ε n2 ,
4
4
√
and therefore α ≤ ε.
Let {x, y} ∈ E(G) be an edges √
whose both endpoints are in A. Suppose
that both x and y have at least εn neighbors in B. Let X1 = {x} and
X2k+1 = {y}. Define X2 = NH (X1 ) and X2k = NH (X2k+1 ) where for a set
X, NH (X) is the set of vertices incident to some vertex in X in the graph H.
For i = 3, 4, · · · , 2k − 1, define Xi+1 = NH (Xi ). We claim that |Xi | ≥ 4εn
for all i. The claim is true for X2 . Now suppose that the claim is true for
Xi and not for Xi+1 . If Xi ⊆ A, then Xi+1 ⊆ B and
eH (Xi , B \ Xi+1 ) = 0.
Therefore
eH (A, B) ≤ |A||B| − |Xi | · |B \ Xi+1 |
√
1 2 √
1
− ε n2
≤ n − εn · (|B| − εn) ≤
4
4
which contradicts√(1). Similar analysis works when Xi ⊆ B.
Hence |Xi | ≥ εn for all i = 2, 3, · · · , 2k. Suppose that there are no
edges between X2k−1 and X2k . Then
1
eH (A, B) ≤ |A||B| − |X2k−1 |X2k | ≤
− ε n2
4
LECTURE 4. REGULARITY LEMMA II
3
and contradicts (1). Hence there exists a cycle of length 2k + 1 containing
both x and y. Therefore for
y} ∈ E(G) whose both endpoints
√ each edge {x, √
are in A, either dH (x) < εn or dH (y) < εn.
a set A0 ⊆ A that intersects all edges in A and dH (a) <
√ Hence we can find
0
εn for all
√ a ∈ A . Since A ∪ B is a MAXCUT of G, for each vertex a ∈ A,
we have εn ≥ |N (a) ∩ B| ≥ |N (a) ∩ A| (otherwise we can move a to B to
obtain a larger cut). Hence
√
eG (A) ≤ |A0 | · εn
√
Also since each vertex a ∈ A0 satisfies dH (a) < εn, we see that
√ n
eG (A, B) ≤ |A||B| − |B| − εn · |A0 | ≤ |A||B| − |A0 |.
4
0
Similarly,
√ all edges in B and
√ we can find a set B ⊆ B that intersects
dH (b) < εn for all b ∈ B 0 . Moreover eG (B) ≤ |B 0 | εn and eG (A, B) ≤
|A||B| − n4 |B 0 |.
Hence
|E(G)| ≤ eG (A) + eG (B) + eG (A, B)
√
1 n
|A0 | + |B 0 |
≤ |A0 | + |B 0 | · εn + |A||B| − ·
2 4
≤ |A||B|.
Thus we see that |A0 | + |B 0 | = 0 and e(G) ≤
n2
4 .
A graph G is color critical if for every edge e ∈ E(G), the graph G \ e
(the graph obtained by removing e) satisfies χ(G \ e) < χ(G). Note that all
odd cycles are color critical. The following theorem extends Theorem 3 to
color critical graphs.
Theorem 4. (Simonovits) If H is a color critical graph, then for all sufficiently large n,
ex(n, H) = e(Tn,χ(H)−1 ),
where Tn,χ(H)−1 is the (χ(H) − 1)-partite Turán graph on n vertices.
Proof sketch. Define r = χ(H) − 1. For simplicity, assume that n is divisible
by r. Let G be a graph on n vertices with at least (1 − 1r ) n2 edges. Let
V = V1 ∪ V2 ∪ · · · ∪ Vr be a r-partition of G which maximizes the number of
edges across the partition. An extension of Theorem 3 can be used to show
that the number of edges across the partition is at least (1 − 1r − ε) n2 edges.
Claim 1. If xy is an edge whose both endpoints are in V1 , then there exists
an index i > 1 such that x or y has at most ε0 n neighbors in Vi .
We omit the proof of the claim. They key idea is that for all subsets
Xi ⊆ Vi and Xj ⊆ Vj of sizes |Xi | ≥ ε0 n and |Xj | ≥ ε0 n, the number of edges
between Xi and Xj is at least |Xi ||Xj | − εn2 . Claim 1 implies the following
claim.
LECTURE 4. REGULARITY LEMMA II
4
Claim 2. For each index i, there exists a set Wi ⊆ Vi that (i) intersects all
edges of G inside Vi , (ii) eG (Vi ) ≤ |Wi | ·ε0 n, and (iii) the number of edges
n
across the partition is at most (1 − 1r ) n2 − 2r
|Wi |.
It easily follows that Wi = ∅ for all i.
2. Weak regularity lemma
Recall that the bound on the number of parts in the regularity lemma
is rather poor. In this section, we study variants of the regularity lemma
where the definition of regularity is weaker but the bound on the number of
parts is dramatically better.
2.1. Frieze and Kannan. The weak regularity lemma of Frieze and Kannan is based on the following definition of regularity.
Definition 5. Let G be a graph on n vertices. A partition Π = V0 ∪ V1 ∪
· · · ∪ Vk is weak ε-regular if |V0 | ≤ εn, Π is an equitable partition, and for
every pair of disjoint subsets X, Y of sizes |X| ≥ εn and |Y | ≥ εn,
X
d(Vi , Vj ) · |X ∩ Vi | · |Y ∩ Vj | ≤ εn2 .
(2)
e(X, Y ) −
i6=j
The quantity above measures the difference between the number of edges
between X and Y and the expected number of edges based on the densities
across the pairs of parts of the partition. The following theorem is the weak
regularity lemma of Frieze and Kannan.
Theorem 6. For all t and ε, there exists T and n0 such that the following
holds for all n ≥ n0 . Every n-vertex graph admits a weak ε-regular partition
2
into k parts for some t ≤ k ≤ T where T ≤ 2−Ω(ε ) .
2
The bound 2−Ω(ε ) is best possible. The proof of Theorem 6 is similar
to the proof of the original regularity lemma. The key difference is that
the number of parts do not increase exponentially. Following is the main
ingredient in the proof of the weak regularity lemma.
Lemma 7. Let Π = (Vi )ki=0 be an equitable partition satisfying |V0 | ≤ εn
that is not weak ε-regular. Then there exists a partition Π0 such that
d2 (Π0 ) ≥ d2 (Π) + ε2 .
Moreover |Π0 | ≤ 3k.
The main gain compared to the original regularity lemma comes from the
fact that we no longer need to take the common refinement of partitions.
Frieze and Kannan developed the weak regularity lemma with algorithmic
applications in mind. In fact for those applications we need the following
slightly different form of the lemma.
LECTURE 4. REGULARITY LEMMA II
5
Theorem 8. For every positive real ε and graph G, there exists a family of
k triples (Si , Ti , di ) where Si , Ti are disjoint subsets of vertices and di is a
real number, such that for every disjoint sets X, Y ⊆ V (G),
k
X
di |Si ∩ X||Ti ∩ Y | ≤ εn2 ,
e(X, Y ) −
i=1
where k ≤
1/ε2 .
Note that di are not necessarily positive nor bounded. The best way to
prove the theorem above is to go through matrices.
I will discuss one application. The MAXCUT problem takes a graph G as
input and asks to determine the largest bipartite subgraph of G. Let OPT be
the number of edges in the largest bipartite subgraph. MAXCUT is known
to be an NP-complete problem and thus one cannot compute OPT exactly
in polynomial time. On the other hand, it is easy to produce a partition
which has at least 12 OPT edges across the partition. A breakthrough result
of Goemans and Williamson gives a polynomial time approximation that
produces a partition which has at least 0.878OPT edges across. Frieze and
Kannan proved the following theorem.
Theorem 9. For every δ there exists a constant c such that the following
holds for every ε. For every n-vertex graph G having at least δn2 edges,
one can find a partition V = V1 ∪ V2 of its vertex set for which e(V1 , V2 ) ≥
2
OPT − εn2 in time 2O(1/ε ) .
Note that the time does not depend on the size of the input.
2.2. Duke, Lefmann, and Rödl. Given a k-partite graph G = (V, E) with
k-partition V = V1 ∪ V2 ∪ · · · ∪ Vk , consider a partition K of V1 × · · · × Vk
into cylinders K = W1 × · · · × Wk where Wi ⊆ Vi for i = 1, 2, · · · , k. Define
Vi (K) = Wi for all i = 1, 2, · · · , k. A cylinder K is ε-regular if all the k2
pairs of subsets (Wi , Wj ) for 1 ≤ i < j ≤ k are ε-regular. The partition
K is cylinder-ε-regular if all but an ε-fraction of the k-tuples (v1 , · · · , vk ) ∈
V1 × · · · × Vk are in ε-regular cylinders in the partition K.
The weak regularity lemma of Duke, Lefmann, and Rödl is as follows.
2 −5
Theorem 10. Let 0 < ε < 12 and β = εk ε . Suppose that G = (V, E) is
a k-partite graph with k-partition V = V1 ∪ · · · ∪ Vk . Then there exists an
cylinder-ε-regular partition K of V1 × · · · × Vk into at most β −1 parts such
that for each K ∈ K and i ∈ [k], we have |Vi (K)| ≥ β|Vi |.
Using this form of the regularity lemma, they proved the following result.
Theorem 11. For each graph H and positive real ε, if n is sufficiently large,
then for every n-vertex graph G one can approximate
the number of copies
of H in G with additive error at most ε nk in time O(n3 ).
LECTURE 4. REGULARITY LEMMA II
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The actual result is stronger than what is stated here. See [2] for the
original statement. The proof of Theorem 11 is quite straightforward once
we have an ‘algorithmic version’ of the Duke-Lefmann-Rödl weak regularity lemma. Here is the proof strategy assuming the algorithmic version.
Suppose that H is a graph on k vertices and G be a graph on n vertices.
Suppose that G contains M copies of H. Denote the vertex set of H as
[k]. Take a random k-partition V1 ∪ · · · ∪ Vk of G. Standard estimates show
that with high probability the number of copies of H that have vertex i in
Vi will be k1k M ± εnk . Hence it suffices to estimate the number of copies
of H where vertex i embeds to vertex Vi . Now apply the weak regularity
lemma to V1 ∪ V2 ∪ · · · ∪ Vk to obtain a cylinder-ε-regular partition K of
V1 × · · · × Vk . For each K ∈ K, we can easily estimate the number of copies
of H having vertex i in Vi (K) (it will be the product of ‘relevant’ densities).
By summing up all these estimates, one can obtain the desired estimate on
M.
3. Strong regularity lemma
Definition 12. For an equitable partition A = {Vi | 1 ≤ i ≤ k} of V (G) and
an equitable refinement B = {Vi,j | 1 ≤ i ≤ k, 1 ≤ j ≤ `} of A, we say that
B is ε-close to A if the following is satisfied. All pairs 1 ≤ i ≤ i0 ≤ k but at
most εk 2 satisfy the following: for all but at most ε`2 of pairs j, j 0 satisfying
1 ≤ j, j 0 ≤ `, we have |d(Vi , Vi0 ) − d(Vi,j , Vi0 ,j 0 )| < ε.
The strong regularity lemma of Alon, Fischer, Krivelevich, and Szegedy
states the following.
Theorem 13. For every function f : N → (0, 1), there exists an integer
S = S(f ) with the following property. For every graph G = (V, E), there is
an equitable partition A of the vertex set V and an equitable refinement B
of A with |B| ≤ S such that the partition A is f (1)-regular, the partition B
is f (|A|)-regular, and B is f (1)-close to A.
A crucial feature is that the regularity parameter of B depends on the
number of parts of A. This provides a more fine-tuned control.
For ‘reasonable’ functions f , the upper bound on S(f ) is at least wowzer
in a power of the inverse of ε = f (1). Recall that the tower function is
defined by T (1) = 2 and T (n) = 2T (n−1) . The wowzer-type function is
defined by W (1) = 2 and W (n) = T (W (n − 1)).
The strong regularity lemma was developed to prove the induced graph
removal lemma. For a graph H and a natural number k, we say that a graph
G is k-far from being induced H-free if at least k edges has to be added to
or deleted from G to make it induced H-free.
Theorem 14. (Induced graph removal lemma) For any graph H on h vertices and ε > 0, there exists δ = δ(ε, H) > 0 such that if a graph G on n
vertices is εn2 -far from being induced H-free, then it contains at least δnh
induced copies of H.
LECTURE 4. REGULARITY LEMMA II
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One can try to mimic the proof of the removal lemma based on the regularity lemma, but there are many difficulties that comes up. The main
lemma used in overcoming this difficulty was proved using the strong regularity lemma.
Theorem 15. For each 0 < ε < 13 , there exists δ = δ(ε) such that every
sufficient large graph G = (V, E) has an equitable partition V = V1 ∪ · · · ∪ Vk
and vertex subsets Wi ⊆ Vi such that |Wi | ≥ δ|V |, each pair (Wi , Wj ) with
1 ≤ i ≤ j ≤ k is ε-regular, and all but at most εk 2 pairs 1 ≤ i ≤ j ≤ k
satisfy |d(Vi , Vj ) − d(Wi , Wj )| ≤ ε.
Note that in the statement above, all pairs (Wi , Wj ) are f (k)-regular
(even for i = j). So far we have been only discussing the regularity of
distinct pairs. For a set X ⊆ V (G), we say that (X, X) is ε-regular if for
every pair of disjoint subsets A ⊆ X and B ⊆ X of sizes |A| ≥ ε|X| and
|B| ≥ ε|Y |, we have |d(A, B) − d(X, X)| ≤ ε. It is not too difficult to
prove the induced removal lemma using Theorem 15 (one must develop the
induced counting lemma).
Proof of Theorem 15 (weak version). We prove a version of Theorem 15 where
we do not impose the pairs (Wi , Wi ) to be ε-regular.
Define f (k) = kε2 for all k. Apply Theorem 13 to find an equitable
partition A and an equitable refinement B with |B| ≤ S such that the
partition A is ε-regular, the partition B is f (|A|)-regular, and B is ε-close
to A. Let A = {Vi | 1 ≤ i ≤ k} and B = {Vi,j | 1 ≤ i ≤ k, 1 ≤ j ≤ `}.
Call a function σ : [k] → [`] bad if (i) there exists a pair (Vi,σ(i) , Vi0 ,σ(i0 ) )
that is not f (k)-regular or (ii) there exists at least 2εk 2 pairs (i, i0 ) such that
|d(Vi , Vi0 ) − d(Vi,σ(i) , Vi0 ,σ(i0 ) )| ≥ ε. The number of functions that are bad
because of (i) is at most
k
· f (k)`2 · `k−2 < ε · `k .
2
On the other hand, if there are M functions that are bad because of (ii),
then
k
2
2
k
M · 2εk ≤ εk · ` +
· ε`2 · `k−2 ,
2
and thus M ≤ 34 `k . Hence there exists at least one function σ that is not
bad. Define Wi = Vi,σ(i) for all i ∈ [k] and note that Wi has the desired
properties.
To prove the general case, one can instead define Wi = Vi,a1 ∪ Vi,a2 ∪ · · · ∪
Vi,at for appropriately chosen indices a1 , a2 , · · · , at .
References
[1] D. Conlon and J. Fox, Bounds for graph regularity and removal lemmas.
[2] R. Duke, H. Lefmann, and V. Rödl, A fast approximation algorithm for computing
the frequencies of subgraphs in a given graph.