The typical structure of maximal triangle-free graphsorion.math.iastate.edu/dept/seminar/slides/20160405-pe... · 2016-04-05 · Lower bound Question (Erd}os) Determine the number

The typical structure ofmaximal triangle-free graphs

Jozsef Balogh, Hong Liu, Sarka Petrıckova, Maryam Sharifzadeh

University of Illinois Urbana-Champaign, University of Warwick

ISU Discrete Mathematics Seminar 2016April 5, 2016

Sarka Petrıckova The typical structure of maximal triangle-free graphs

History

Theorem (Mantel 1907)

The maximum number of edges in an n-vertex triangle-free graph is n2/4.

⇒ The number of labeled n-vertex triangle-free graphs is at least 2n2/4.

Theorem (Erdos-Kleitman-Rothschild 1976)

The number of triangle-free graphs on [n] := {1, . . . , n} is 2n2/4+o(n2).Moreover, almost all triangle-free graphs are bipartite.

Most bipartite graphs are not maximal triangle-free.(maximal triangle-free = triangle-free such that adding any edge resultsin a triangle)

Question (Erdos)

Determine or estimate the number of maximal triangle-free graphs on thevertex set [n].


History







Question (Erdos)



History







Question (Erdos)



History






Most bipartite graphs are not maximal triangle-free.

(maximal triangle-free = triangle-free such that adding any edge resultsin a triangle)

Question (Erdos)



History







Question (Erdos)



History







Question (Erdos)



Lower bound

Question (Erdos)

Determine the number of maximal triangle-free graphs on [n].

Lower bound:

Start with a graph on a vertex set X ∪ Y with |X | = |Y | = n/2 s. t.X induces a perfect matching, Y is an independent set.

For each pair of a matching edge x1x2 in X and a vertex y ∈ Y , addexactly one of the edges x1y or x2y .

⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)

n/4 matching edges in X , n/2 vertices in Y ⇒ 2n/4·n/2 = 2n2/8 graphs


Lower bound

Question (Erdos)


Lower bound:



⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Lower bound

Question (Erdos)


Lower bound:



⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Lower bound

Question (Erdos)


Lower bound:



⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Lower bound

Question (Erdos)


Lower bound:



⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)

n/4 matching edges in X , n/2 vertices in Y ⇒ 2n/4·n/2 = 2n2/8 graphsSarka Petrıckova The typical structure of maximal triangle-free graphs

Upper bound

Theorem (Balogh-P.)

The number of maximal triangle-free graphs on the vertex set [n] is

2n2/8+o(n2).

TOOLS:

1 Container Method by Balogh-Morris-Samotij; Saxton-Thomason:

There is a family F of ≤ 2O(log n·n3/2) n-vertex graphs such thatevery F ∈ F has o(n3) triangles and ∀G n-vertex triangle-free∃F ∈ F with G ⊆ F .

2 Ruzsa-Szemeredi triangle-removal lemma:Every n-vertex graph G with o(n3) triangles can be madetriangle-free by removing o(n2) edges.

3 Hujter-Tuza: Every triangle-free graph G has at most 2|G |/2

maximal independent sets.


Upper bound

Theorem (Balogh-P.)


2n2/8+o(n2).

TOOLS:1 Container Method by Balogh-Morris-Samotij; Saxton-Thomason:






Upper bound

Theorem (Balogh-P.)


2n2/8+o(n2).

TOOLS:







Upper bound

Theorem (Balogh-P.)


2n2/8+o(n2).

TOOLS:







Upper bound

Theorem (Balogh-P.)


2n2/8+o(n2).

TOOLS:







Upper bound

Theorem (Balogh-P.)


2n2/8+o(n2).

TOOLS:







Upper bound (Proof)

Every maximal triangle-free graph G can be built in the following threesteps:

(S1) Choose a container F ∈ F such that G ⊆ F .

2o(n2)

Find A,B ∈ F such that A ∪ B = F , A is triangle-free, and B haso(n2) edges.

(S2) Choose a triangle-free subgraph S ⊆ B.

2o(n2)

(S3) Extend S to a max. triangle-free graph by adding some edges of A.(= choose a subgraph A′ ⊆ A such that A′ ∪S is max. triangle-free.)

will show ≤ 2n2/8


Upper bound (Proof)



2o(n2)



2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)



2o(n2)



2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)



2o(n2)



2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)


(S1) Choose a container F ∈ F such that G ⊆ F . 2o(n2)



2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)




(S2) Choose a triangle-free subgraph S ⊆ B. 2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)




(S2) Choose a triangle-free subgraph S ⊆ B. 2o(n2)


will show ≤ 2n2/8


Upper bound (Proof)

WLOG, there are no triangles with two edges in S and one in A.

Given triangle-free graphs A and S , define link graph L of S on A:

V (L) := E (A)E (L) := {a1a2 : ∃s ∈ E (S) s.t. {a1, a2, s} forms a triangle}

C1: L is triangle-free.

A′ ∪ S maximal triangle-free ⇒ E (A′) maximal independent set in L.

C2: #maximal triangle-free graphs of A ∪ S containing S is ≤ MIS(L).

#extensions in (S3)C2≤ MIS(L)

C1&Hujter-Tuza≤ 2|L|/2 ≤ 2n2/8

Together:#maximal triangle-free graphs on [n] ≤ 2o(n2) · 2o(n2) · 2n2/8 = 2n2/8+o(n2).

�


Upper bound (Proof)

WLOG, there are no triangles with two edges in S and one in A.Given triangle-free graphs A and S , define link graph L of S on A:








�


Upper bound (Proof)









�


Upper bound (Proof)









�


Upper bound (Proof)









�


Upper bound (Proof)






#extensions in (S3)

C2≤ MIS(L)



�


Upper bound (Proof)









�


Upper bound (Proof)







C1&Hujter-Tuza≤ 2|L|/2

≤ 2n2/8


�


Upper bound (Proof)









�


Upper bound (Proof)









�


Upper bound (Proof)









�


Structure of maximal H-free graphs

Typical structure of H-free graphs has been studied when:

H is a large clique(Balogh-Bushaw-Collares Neto-Liu-Morris-Sharifzadeh,Balogh-Morris-Samotij-Warnke),

H is a fixed color-critical subgraph (Promel-Steger),

H is a finite family of subgraphs (Balogh-Bollobas-Simonovits),

H is an induced subgraph (Balogh-Butterfield).

Many more results for sparse graphs, hypergraph, and other discretestructures.

Almost nothing known about the structure of maximal H-free graphs!


Structure of maximal triangle-free graphs

Triangle-free:

Number: 2n2/4+o(n2)

Structure: almost all bipartite

Maximal Triangle-free:

Number: 2n2/8+o(n2)

Structure: ???????

Theorem (Balogh, Liu, P., Sharifzadeh 2015+)

Almost every maximal triangle-free graph G admits a vertex partitionX ∪Y such that G [X ] is a perfect matching and Y is an independent set.

I.e. almost all maximal triangle-free graphs have the same structure asthe graphs in the lower bound construction!

⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Triangle-free:

Number: 2n2/4+o(n2)



Number: 2n2/8+o(n2)

Structure: ???????




⇒

X YX Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

x1

x2

NY (x1)

NY (x2)



Theorem (Balogh-Liu-P.-Sharifzadeh)


TOOLS:

1 Container Method

2 Triangle-removal lemma

3 Erdos-Simonovits stability theorem: Every triangle-free graph can bemade bipartite by removing o(n2) edges.

4 Generalization of Hujter-Tuza: Every triangle-free graph G with atleast k vertex-disjoint P3’s has at most 2|G |/2−k/25 maximalindependent sets.





TOOLS:1 Container Method2 Triangle-removal lemma3 Erdos-Simonovits stability theorem: Every triangle-free graph can be

made bipartite by removing o(n2) edges.4 Generalization of Hujter-Tuza: Every triangle-free graph G with at

least k vertex-disjoint P3’s has at most 2|G |/2−k/25 maximalindependent sets.





TOOLS:

1 Container Method








TOOLS:

1 Container Method








TOOLS:

1 Container Method








TOOLS:

1 Container Method








TOOLS:

1 Container Method






The proof has 2 main parts:

1 Asymptotic result: a.a. maximal triangle-free graphs have astructure close to the desired one.

X Y

X ′ Y ′

o(n)

2 Precise result: there are exponentially fewer ‘bad’ graphs (maximaltriangle-free graphs without the desired strucure).


1. Proof of Asymptotic result

Lemma

Almost every maximal triangle-free graph G satisfies the following: forevery max-cut X ∪ Y there exist X ′ ⊆ X and Y ′ ⊆ Y such that:

X − X ′ induced a perfect matching and X ′ = o(n), and

Y − Y ′ is an independent set and Y ′ = o(n).

X Y

X ′ Y ′

o(n)



Lemma

Almost every maximal triangle-free graph G satisfies the following: forevery max-cut X ∪ Y there exist X ′ ⊆ X and Y ′ ⊆ Y such that:

X − X ′ induced a perfect matching and X ′ = o(n), and

Y − Y ′ is an independent set and Y ′ = o(n).

X Y

X ′ Y ′

o(n)



Every maximal triangle-free graph G can be built in these three steps:

(S1) Choose a max-cut X ∪ Y for G .

2n

(S2) Choose triangle-free graphs S and T on the vertex sets X and Y ,respectively.

a. a. max. triangle-free graphs are o(n2)-close to bipartite

2o(n2)

(S3) Extend S ∪ T to a maximal triangle-free graph by adding someedges between X and Y .

???

X Y





2n



2o(n2)


???

X Y





2n



2o(n2)


???

X Y





2n



2o(n2)


???

X Y





2n



2o(n2)


???

X Y




(S1) Choose a max-cut X ∪ Y for G . 2n



2o(n2)


???

X Y







2o(n2)


???

X Y

S T







2o(n2)


???

X Y

S T






a. a. max. triangle-free graphs are o(n2)-close to bipartite 2o(n2)


???

X Y

S T








???

X Y

S T







(S3) Extend S ∪ T to a maximal triangle-free graph by adding someedges between X and Y . ???

X Y

S T



Let A be the complete bipartite graph on parts X and Y .Define link graph L := LS∪T [A] of S ∪ T on A:

V (L) := E (A)E (L) := {a1a2 : ∃e ∈ E (S ∪ T ) s.t. {a1, a2, e} forms a triangle}


C2: The number of maximal triangle-free subgraphs of S ∪ A containingS is equal to MIS(L).

C3: L = S�T .

Generalization of Hujter-Tuza + C1 imply:

If L contains at least k vertex-disjoint P3’s, then MIS(L) ≤ 2|L|/2−k/25.




V (L) := E (A)

E (L) := {a1a2 : ∃e ∈ E (S ∪ T ) s.t. {a1, a2, e} forms a triangle}



C3: L = S�T .









C3: L = S�T .









C3: L = S�T .









C3: L = S�T .









C3: L = S�T .









C3: L = S�T .





IDEA: If S ∪ T does not have the desired structure, then we find manyP3’s in L = S�T , which means that the number of such maximaltriangle-free graphs is small.

In almost all maximal triangle-free graphs:

1 S ∪ T is mostly a union of a matching and an independent set.

Suppose there are cn vertex-disjoint P3’s in S .Then there are cn|T | vertex-disjoint P3’s in L = S�T .Then MIS(L) ≤ 2|S||T |/2−cn|T |/25 - small!.Since MIS(L) is exactly the number of extensions in (S3), the familyof maximal triangle-free graphs with such S and T is negligible.

2 Only one of S and T (say S) can have a matching of linear size.

3 There are only o(n) isolated vertices in S .






































Suppose there are cn vertex-disjoint P3’s in S .

Then there are cn|T | vertex-disjoint P3’s in L = S�T .Then MIS(L) ≤ 2|S||T |/2−cn|T |/25 - small!.Since MIS(L) is exactly the number of extensions in (S3), the familyof maximal triangle-free graphs with such S and T is negligible.








Suppose there are cn vertex-disjoint P3’s in S .Then there are cn|T | vertex-disjoint P3’s in L = S�T .

Then MIS(L) ≤ 2|S||T |/2−cn|T |/25 - small!.Since MIS(L) is exactly the number of extensions in (S3), the familyof maximal triangle-free graphs with such S and T is negligible.








Suppose there are cn vertex-disjoint P3’s in S .Then there are cn|T | vertex-disjoint P3’s in L = S�T .Then MIS(L) ≤ 2|S||T |/2−cn|T |/25 - small!.

Since MIS(L) is exactly the number of extensions in (S3), the familyof maximal triangle-free graphs with such S and T is negligible.












2. Proof of Exact result

Fix X ,Y ,M. Suppose |X | is even.

Good graphs:

G – all maximal triangle-free graphs with max-cut X ∪ Y ,G [X ] = M, and Y an independent set.

Bad graphs:

B(s, t) – all maximal triangle-free graphs with max-cut X ∪ Y , withmaximum matching M ′ ⊆ M covering all but o(n) vertices in X , svertex-disj. P3’s in S = G [X ] and matching of size t in t = G [Y ].

B(r) – subclass of B(0, 0) with exactly r isolated vertices in S .

Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.



Fix X ,Y ,M. Suppose |X | is even.Good graphs:


Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.





Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.





Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.





Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.





Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.





Bad graphs:



Need to show:

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)

the number s of vertex-disjoint P3’s in S as well as the size t ofmaximum matching in T is o(n)

Using similar techniques as before plus the above assumptions:

If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)



If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)



If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)



If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)



If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.



We can assume that:

|X |, |Y | ≥ n/2− o(n)

∆(X ),∆(Y ) = o(n)



If s + t ≥ 1, then B(s, t) ≤ 2|X ||Y |/2−n/200.

If r ≥ 1, then B(r) ≤ 2|X ||Y |/2−n/6.

|⋃s+t≥1 B(s, t) ∪⋃r≥1 B(r)||G| ≤ (n2 + n)2|X ||Y |/2−n/200

(1 + o(1))2|X ||Y |/2≤ 2−n/300.


Kr+1-free graphs

Turan number t(n, r) =(1− 1

r

)n2

2


The number of Kr+1-free graphs on [n] is 2t(n,r)+o(n2).

Moreover, almost all Kr+1-free graphs are r -partite(Kolaitis-Promel-Rothschild 1987).

Theorem (Erdos-Frankl-Rodl 1986)

If χ(H) ≥ 3, then the number of H-free graphs on [n] is 2t(n,χ(H)−1)+o(n2).

+ many more structural results on H-free graphs for various graphs(families of graphs) H.


Kr+1-free graphs


r

)n2

2








Kr+1-free graphs


r

)n2

2








Maximal Kr+1-free graphs

What about the number mr+1(n) of maximal Kr+1-free graphs on [n]?

A discussion with Alon and Luczak led to the following construction:

X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges

⇒ mr+1(n) ≥ 2t(n,r)/2+o(n2)

Upper bound: Erdos-Kleitman-Rothschild: ⇒ mr+1(n) ≤ 2t(n,r)+o(n2)

Improvement: ∀r ≥ 3 ∃εr > 0: mr+1(n) ≤ 2t(n,r)−εrn2

(n suff. large)





X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges

⇒ mr+1(n) ≥ 2t(n,r)/2+o(n2)



(n suff. large)





X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges

⇒ mr+1(n) ≥ 2t(n,r)/2+o(n2)



(n suff. large)


Open problems

Is the lower bound the “correct value”? Is it true that

mr+1(n) = 2t(n,r)/2+o(n2) ?

What can we say about the typical structure of maximal Kr+1-freegraphs for r ≥ 3? Do almost all graphs look like the graphs fromlower bound construction, as in the triangle-case?

X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges


Open problems


mr+1(n) = 2t(n,r)/2+o(n2) ?

What can we say about the typical structure of maximal Kr+1-freegraphs for r ≥ 3?

Do almost all graphs look like the graphs fromlower bound construction, as in the triangle-case?

X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges


Open problems


mr+1(n) = 2t(n,r)/2+o(n2) ?

What can we say about the typical structure of maximal Kr+1-freegraphs for r ≥ 3? Do almost all graphs look like the graphs fromlower bound construction, as in the triangle-case?

X Y

x1

x2

y

Add 1edgefor

everypair

(x1x2, y)

X1 Y

X2

Add 1 edge

Add 3 edges


Thank you.


Documents

The typical structure of maximal triangle-free graphsorion.math.iastate.edu/dept/seminar/slides/20160405-pe... · 2016-04-05 · Lower bound Question (Erd}os) Determine the number