Semidefinite method and Caccetta-H aggvist conjecture Jan Volec - - PowerPoint PPT Presentation

Semidefinite method and Caccetta-H¨ aggvist conjecture

Jan Volec

ETH Z¨ urich

joint work with Jean-S´ ebastien Sereni and R´ emi De Joannis De Verclos

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle.

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle. 3k + 1 vertices, connect each vertex i → i + 1, i + 2, . . . , i + k

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle. 3k + 1 vertices, connect each vertex i → i + 1, i + 2, . . . , i + k

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle. G and H extremal graphs − → G × H lexicographic product

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle. G and H extremal graphs − → G × H lexicographic product

Caccetta-H¨ aggvist conjecture

Conjecture (Caccetta-H¨ aggvist, 1978)

Every n-vertex digraph with minimum out-degree at least k contains an oriented cycle of length at most ⌈n/k⌉.

Theorem (Shen, 2000)

C-H conjecture holds for k ≤

• n/2.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle. G and H extremal graphs − → G × H lexicographic product

Triangle case

Every n-vertex oriented graph with minimum out-degree at least n/3 contains an oriented triangle.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least c · n contains an oriented triangle.

Triangle case

Every n-vertex oriented graph with minimum out-degree at least c · n contains an oriented triangle.

• Caccetta-H¨

aggkvist (1978): c < (3 − √ 5)/2 ≈ 0.3819

Triangle case

Every n-vertex oriented graph with minimum out-degree at least c · n contains an oriented triangle.

• Caccetta-H¨

aggkvist (1978): c < (3 − √ 5)/2 ≈ 0.3819

• Bondy (1997): c < (2

√ 6 − 3)/5 ≈ 0.3797

Triangle case

Every n-vertex oriented graph with minimum out-degree at least c · n contains an oriented triangle.

• Caccetta-H¨

aggkvist (1978): c < (3 − √ 5)/2 ≈ 0.3819

• Bondy (1997): c < (2

√ 6 − 3)/5 ≈ 0.3797

• Shen (1998): c < 3 −

√ 7 ≈ 0.3542

• Hamburger, Haxell, Kostochka (2007): c < 0.3531

Triangle case

Every n-vertex oriented graph with minimum out-degree at least c · n contains an oriented triangle.

• Caccetta-H¨

aggkvist (1978): c < (3 − √ 5)/2 ≈ 0.3819

• Bondy (1997): c < (2

√ 6 − 3)/5 ≈ 0.3797

• Shen (1998): c < 3 −

√ 7 ≈ 0.3542

• Hamburger, Haxell, Kostochka (2007): c < 0.3531
• Hladk´

y, Kr´ al’, Norin (2009): c < 0.3465

• Razborov (2011): if D is {F1, F2, F3}-free, then C-H holds

F1 F2 F3

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F
• limit object – function q: finite T-free or.graphs F → [0, 1]

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F
• limit object – function q: finite T-free or.graphs F → [0, 1]
• q yields homomorphism from linear combinations of F to R

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F
• limit object – function q: finite T-free or.graphs F → [0, 1]
• q yields homomorphism from linear combinations of F to R
• the set of limit objects LIM = {homomorphism q : q(F) ≥ 0}

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F
• limit object – function q: finite T-free or.graphs F → [0, 1]
• q yields homomorphism from linear combinations of F to R
• the set of limit objects LIM = {homomorphism q : q(F) ≥ 0}
• semidefinite method: relaxing optimization problems on LIM

Flag Algebras and Semidefinite Method

• developed by Razborov (2010), closely related to graph limits
• consider sequence of T-free oriented graphs G1, G2, . . .
• pk(F) := probability that random |F| vertices of Gk induces F
• always has a subsequence s.t. values pk(F) converge for all F
• sequence (Gk) is convergent if pk(F) converge for all F
• limit object – function q: finite T-free or.graphs F → [0, 1]
• q yields homomorphism from linear combinations of F to R
• the set of limit objects LIM = {homomorphism q : q(F) ≥ 0}
• semidefinite method: relaxing optimization problems on LIM
• we optimize on LIMEXT = {q ∈ LIM : q is extremal for C-H}

Flag Algebras – basic properties of q

• linear extension of q:

q

• α1 ×

+ α2 ×

• := α1 · q
• + α2 · q

Flag Algebras – basic properties of q

• linear extension of q:

q

• α1 ×

+ α2 ×

• := α1 · q
• + α2 · q
• lifting up in q:

q = q 1 3 × + 2 3 × + 2 3 × + 2 3 × +

Flag Algebras – basic properties of q

• linear extension of q:

q

• α1 ×

+ α2 ×

• := α1 · q
• + α2 · q
• lifting up in q:

q = q 1 3 × + 2 3 × + 2 3 × + 2 3 × +

• product of graphs in q:

q ·q( ) = q 1 3 × + 2 3 × + 2 3 × + 2 3 × +

Flag Algebras – basic properties of q

• linear extension of q:

q

• α1 ×

+ α2 ×

• := α1 · q
• + α2 · q
• lifting up in q:

q = q 1 3 × + 2 3 × + 2 3 × + 2 3 × +

• product of graphs in q:

q ·q( ) = q 1 3 × + 2 3 × + 2 3 × + 2 3 × +

• =

⇒ we define × := 1 3 × + 2 3 × + 2 3 × + 2 3 × +

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

Gk σ =

2 1

F σ =

1 2 1 2

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

Gk σ =

2 1

F σ =

1 2 1 2

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

Gk σ =

2 1

F σ =

1 2 1 2

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

Gk σ =

2 1

F σ =

1 2 1 2

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

• rand. functions pσ

k weakly converge to a random function qσ

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

• rand. functions pσ

k weakly converge to a random function qσ

• furthermore, q uniquely determines qσ for every σ

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

• rand. functions pσ

k weakly converge to a random function qσ

• furthermore, q uniquely determines qσ for every σ
• Averaging argument:

q = 2 · E1q1

1

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

• rand. functions pσ

k weakly converge to a random function qσ

• furthermore, q uniquely determines qσ for every σ
• Averaging argument:

q = 2 · E1q1

1

• =

⇒ define

• 1
• 1 := 1

2 ×

Flag Algebras – rooted homomorphisms

• let σ be a graph with pk(σ) > 0 for all sufficiently large k’s
• for a fixed copy C of σ in Gk, consider probabilities pC

k (F σ)

F σ – a graph with fixed embedding of σ → function pC

k

• a random copy C of σ in Gk → a random function pσ

k

• rand. functions pσ

k weakly converge to a random function qσ

• furthermore, q uniquely determines qσ for every σ
• Averaging argument:

q = 2 · E1q1

1

• =

⇒ define

• 1
• 1 := 1

2 ×

• Cauchy-Schwarz inequality:

αFF 2

σ ≥

αFF

• σ

2 ≥ 0

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2.

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

1 3 q

• ≥ q

2

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

1 3 q

• ≥ q

2

q = q

• 1

3 + 2 3 +

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

1 3 q

• ≥ q

2

q = q

• 1

3 + 2 3

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

1 3 q

• ≥ q

2

q = q

• 1

3 + 2 3 q ≥ 2 3 q

• ≥ 2q

2

Flag Algebras – Mantel’s Theorem

If (Gn) conv. sequence of triangle-free graphs, then q ≤ 1

2. 1

• 2
• 1

q

• ≥

1

• 1

q

• 2

1 3 q

• ≥ q

2

q = q

• 1

3 + 2 3 q ≥ 2 3 q

• ≥ 2q

2

1 2 ≥q

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

• can be written in flag algebras language as f σ ≥ 0 for σ = uv

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

• can be written in flag algebras language as f σ ≥ 0 for σ = uv
• for every σ-flag F σ we have
• F σ × f σ

σ ≥ 0

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

• can be written in flag algebras language as f σ ≥ 0 for σ = uv
• for every σ-flag F σ we have
• F σ × f σ

σ ≥ 0

• but also for every αFF σ it holds
• ( αFF σ)2 × f σ

σ ≥ 0

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

• can be written in flag algebras language as f σ ≥ 0 for σ = uv
• for every σ-flag F σ we have
• F σ × f σ

σ ≥ 0

• but also for every αFF σ it holds
• ( αFF σ)2 × f σ

σ ≥ 0

Theorem

Every n-vertex oriented graph with minimum out-degree at least 0.3386n contains an oriented triangle.

Obtaining the bound

Use SDP for generating usefull Cauchy-Schwarz inequalities Inductive arguments

• Suppose G is an extremal example for C-H with δ+(G) = c
• edge uv: |N+(v)| + |N−(u) ∪ N−(v)| + (1 − c)|N+(u) ∩ N+(v)| ≤ n

(otherwise G[N+(u) ∩ N+(v)] has larger minimum δ+ than G)

• can be written in flag algebras language as f σ ≥ 0 for σ = uv
• for every σ-flag F σ we have
• F σ × f σ

σ ≥ 0

• but also for every αFF σ it holds
• ( αFF σ)2 × f σ

σ ≥ 0

Theorem

Every n-vertex oriented graph with minimum out-degree at least 0.3386n contains an oriented triangle.

Thank you for your attention!