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].
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].
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].
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].
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].
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].
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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 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.
Sarka 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.
Sarka 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.
Sarka 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.
Sarka 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.
Sarka 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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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).
�
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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!
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
Theorem (Balogh-Liu-P.-Sharifzadeh)
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.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Structure of maximal triangle-free graphs
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).
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
S T
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
S T
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
S T
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
S T
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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
S T
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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}
C1: L is triangle-free.
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
1. Proof of Asymptotic result
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 .
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
2. Proof of Exact result
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Kr+1-free graphs
Turan number t(n, r) =(1− 1
r
)n2
2
Theorem (Erdos-Kleitman-Rothschild 1976)
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Kr+1-free graphs
Turan number t(n, r) =(1− 1
r
)n2
2
Theorem (Erdos-Kleitman-Rothschild 1976)
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Kr+1-free graphs
Turan number t(n, r) =(1− 1
r
)n2
2
Theorem (Erdos-Kleitman-Rothschild 1976)
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.
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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)
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
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
Sarka Petrıckova The typical structure of maximal triangle-free graphs
Thank you.
Sarka Petrıckova The typical structure of maximal triangle-free graphs