Upper bounds on the size of 4 - and 6 -cycle-free subgraphs of the - - PowerPoint PPT Presentation

▶
upper bounds on the size of 4 and 6 cycle free subgraphs
SMART_READER_LITE
LIVE PREVIEW

Upper bounds on the size of 4 - and 6 -cycle-free subgraphs of the - - PowerPoint PPT Presentation

Feb 07, 2023 120 likes •544 views

Upper bounds on the size of 4 - and 6 -cycle-free subgraphs of the hypercube Ping Hu Joint work with J ozsef Balogh, Bernard Lidick y and Hong Liu University of Illinois at Urbana-Champaign MIGHTY LII - April 28, 2012 Ping Hu (UIUC)

slide-1
SLIDE 1

Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube

Ping Hu Joint work with J´

  • zsef Balogh, Bernard Lidick´

y and Hong Liu

University of Illinois at Urbana-Champaign

MIGHTY LII - April 28, 2012

Ping Hu (UIUC) MIGHTY LII 1 / 14

slide-2
SLIDE 2

Hypercube

Qn is n-dimensional hypercube (n-cube)

Q1 Q2 Q3

Ping Hu (UIUC) MIGHTY LII 2 / 14

slide-3
SLIDE 3

Hypercube

Qn is n-dimensional hypercube (n-cube)

Q1 Q2 Q3

e(G) := |E(G)|

Ping Hu (UIUC) MIGHTY LII 2 / 14

slide-4
SLIDE 4

Hypercube

Qn is n-dimensional hypercube (n-cube)

Q1 Q2 Q3

e(G) := |E(G)| exQ(n, F) := the maximum number of edges of a F-free subgraph of Qn

Ping Hu (UIUC) MIGHTY LII 2 / 14

slide-5
SLIDE 5

Hypercube

Qn is n-dimensional hypercube (n-cube)

Q1 Q2 Q3

e(G) := |E(G)| exQ(n, F) := the maximum number of edges of a F-free subgraph of Qn πQ(F) = lim

n→∞

exQ(n, F) e(Qn)

Ping Hu (UIUC) MIGHTY LII 2 / 14

slide-6
SLIDE 6

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2

Ping Hu (UIUC) MIGHTY LII 3 / 14

slide-7
SLIDE 7

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2

Q7 Q7

πQ(C4) ≥ 1/2

Ping Hu (UIUC) MIGHTY LII 3 / 14

slide-8
SLIDE 8

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2

Q7 Q7

πQ(C4) ≥ 1/2

Ping Hu (UIUC) MIGHTY LII 3 / 14

slide-9
SLIDE 9

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2

Theorem (Chung [1992], Brouwer–Dejter–Thomassen [1993])

πQ(C6) ≥ 1/4

Ping Hu (UIUC) MIGHTY LII 3 / 14

slide-10
SLIDE 10

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2

Theorem (Chung [1992], Brouwer–Dejter–Thomassen [1993])

πQ(C6) ≥ 1/4

Theorem (Conder [1993])

πQ(C6) ≥ 1/3

Ping Hu (UIUC) MIGHTY LII 3 / 14

slide-11
SLIDE 11

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2.

Theorem (Chung [1992])

πQ(n, C2t) = 0 for even t ≥ 4.

Theorem (F¨ uredi–¨ Ozkahya [2009])

πQ(C2t) = 0 for odd t ≥ 7.

Ping Hu (UIUC) MIGHTY LII 4 / 14

slide-12
SLIDE 12

πQ(C2t)

Conjecture (Erd˝

  • s [1984])

πQ(C4) = 1/2, πQ(C2t) = 0 for t > 2.

Theorem (Chung [1992])

πQ(n, C2t) = 0 for even t ≥ 4.

Theorem (F¨ uredi–¨ Ozkahya [2009])

πQ(C2t) = 0 for odd t ≥ 7. if πQ(C10) = 0 is still open.

Ping Hu (UIUC) MIGHTY LII 4 / 14

slide-13
SLIDE 13

πQ(C4)

Theorem (Brass–Harborth–Nienborg [1995])

exQ(n, C4) ≥ 1

2(1 + 1 √n)e(Qn) (valid when n is a power of 4)

Ping Hu (UIUC) MIGHTY LII 5 / 14

slide-14
SLIDE 14

πQ(C4)

Theorem (Brass–Harborth–Nienborg [1995])

exQ(n, C4) ≥ 1

2(1 + 1 √n)e(Qn) (valid when n is a power of 4)

Theorem (Chung [1992])

πQ(C4) ≤ 0.62284.

Ping Hu (UIUC) MIGHTY LII 5 / 14

slide-15
SLIDE 15

πQ(C4)

Theorem (Brass–Harborth–Nienborg [1995])

exQ(n, C4) ≥ 1

2(1 + 1 √n)e(Qn) (valid when n is a power of 4)

Theorem (Chung [1992])

πQ(C4) ≤ 0.62284.

Theorem (Thomason–Wagner [2009])

πQ(C4) ≤ 0.62256.

Ping Hu (UIUC) MIGHTY LII 5 / 14

slide-16
SLIDE 16

πQ(C4)

Theorem (Brass–Harborth–Nienborg [1995])

exQ(n, C4) ≥ 1

2(1 + 1 √n)e(Qn) (valid when n is a power of 4)

Theorem (Chung [1992])

πQ(C4) ≤ 0.62284.

Theorem (Thomason–Wagner [2009])

πQ(C4) ≤ 0.62083.

Ping Hu (UIUC) MIGHTY LII 5 / 14

slide-17
SLIDE 17

πQ(C4)

Theorem (Brass–Harborth–Nienborg [1995])

exQ(n, C4) ≥ 1

2(1 + 1 √n)e(Qn) (valid when n is a power of 4)

Theorem (Chung [1992])

πQ(C4) ≤ 0.62284.

Theorem (Thomason–Wagner [2009])

πQ(C4) ≤ 0.62083.

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C4) ≤ 0.6068.

Ping Hu (UIUC) MIGHTY LII 5 / 14

slide-18
SLIDE 18

πQ(n, C6)

Theorem (Conder [1993])

πQ(C6) ≥ 1/3.

Ping Hu (UIUC) MIGHTY LII 6 / 14

slide-19
SLIDE 19

πQ(n, C6)

Theorem (Conder [1993])

πQ(C6) ≥ 1/3.

Theorem (Chung [1992])

πQ(C6) ≤ √ 2 − 1 ≈ 0.41421.

Ping Hu (UIUC) MIGHTY LII 6 / 14

slide-20
SLIDE 20

πQ(n, C6)

Theorem (Conder [1993])

πQ(C6) ≥ 1/3.

Theorem (Chung [1992])

πQ(C6) ≤ √ 2 − 1 ≈ 0.41421.

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C6) ≤ 0.3755.

Ping Hu (UIUC) MIGHTY LII 6 / 14

slide-21
SLIDE 21

Flag Algebras

Definition

p(H, G): the probability that a random |V (H)|-set U in V (G) induces G[U] isomorphic to H. Razborov [2007] developed flag algebras. Let G be the family of graphs forbidding some structures, then flag algebras can be used to bound lim

G∈G,|V (G)|→∞ p(H, G).

Ping Hu (UIUC) MIGHTY LII 7 / 14

slide-22
SLIDE 22

Results using Flag Algebras

Ping Hu (UIUC) MIGHTY LII 8 / 14

slide-23
SLIDE 23

Results using Flag Algebras

Theorem (Hladk´ y–Kr´ al’–Norine [2009])

Every n-vertex digraph with minimum outdegree at least 0.3465n contains a triangle.

Ping Hu (UIUC) MIGHTY LII 8 / 14

slide-24
SLIDE 24

Results using Flag Algebras

Theorem (Hladk´ y–Kr´ al’–Norine [2009])

Every n-vertex digraph with minimum outdegree at least 0.3465n contains a triangle.

Theorem (Hatami–Hladk´ y–Kr´ al’–Norine–Razborov [2011], Grzesik [2011])

The number of C5s in a triangle-free graph of order n is at most (n/5)5.

Ping Hu (UIUC) MIGHTY LII 8 / 14

slide-25
SLIDE 25

Results using Flag Algebras

Theorem (Hladk´ y–Kr´ al’–Norine [2009])

Every n-vertex digraph with minimum outdegree at least 0.3465n contains a triangle.

Theorem (Hatami–Hladk´ y–Kr´ al’–Norine–Razborov [2011], Grzesik [2011])

The number of C5s in a triangle-free graph of order n is at most (n/5)5.

Theorem (Falgas-Ravry–Vaughan [2011])

π(K −

4 , C5, F3,2) = 12/49, π(K − 4 , F3,2) = 5/18.

F3,2 : {123, 145, 245, 345}, C5 : {123, 234, 345, 451, 512}.

Ping Hu (UIUC) MIGHTY LII 8 / 14

slide-26
SLIDE 26

Proof by an Example

Example

πQ(C4) ≤ 2/3

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-27
SLIDE 27

Proof by an Example

Example

πQ(C4) ≤ 2/3 Bound infinite problem by a finite piece.

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-28
SLIDE 28

Proof by an Example

Example

πQ(C4) ≤ 2/3 Bound infinite problem by a finite piece.

Definition

Hn: the family of spanning subgraphs of Qn not containing C4. Let H ∈ Hs, G ∈ Hn, s < n, p(H, G) is the probability that a random s-hypercube vertex set in G induces H. ρ(G) = e(G)/e(Qn).

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-29
SLIDE 29

Proof by an Example

Example

πQ(C4) ≤ 2/3 Bound infinite problem by a finite piece.

Definition

Hn: the family of spanning subgraphs of Qn not containing C4. Let H ∈ Hs, G ∈ Hn, s < n, p(H, G) is the probability that a random s-hypercube vertex set in G induces H. ρ(G) = e(G)/e(Qn). ρ(G) =

  • H∈Hs

ρ(H)p(H, G)

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-30
SLIDE 30

Proof by an Example

Example

πQ(C4) ≤ 2/3 Bound infinite problem by a finite piece.

Definition

Hn: the family of spanning subgraphs of Qn not containing C4. Let H ∈ Hs, G ∈ Hn, s < n, p(H, G) is the probability that a random s-hypercube vertex set in G induces H. ρ(G) = e(G)/e(Qn). ρ(G) =

  • H∈Hs

ρ(H)p(H, G) ρ(G) ≤ max

H∈Hs ρ(H)

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-31
SLIDE 31

Proof by an Example

Example

πQ(C4) ≤ 2/3 Bound infinite problem by a finite piece.

Definition

Hn: the family of spanning subgraphs of Qn not containing C4. Let H ∈ Hs, G ∈ Hn, s < n, p(H, G) is the probability that a random s-hypercube vertex set in G induces H. ρ(G) = e(G)/e(Qn). ρ(G) =

  • H∈Hs

ρ(H)p(H, G) πQ(C4) ≤ max

H∈Hs ρ(H)

Ping Hu (UIUC) MIGHTY LII 9 / 14

slide-32
SLIDE 32

Is the bound good?

ρ(G) =

  • H∈Hs

ρ(H)p(H, G) πQ(C4) ≤ max

H∈Hs ρ(H)

Ping Hu (UIUC) MIGHTY LII 10 / 14

slide-33
SLIDE 33

Is the bound good?

ρ(G) =

  • H∈Hs

ρ(H)p(H, G) πQ(C4) ≤ max

H∈Hs ρ(H)

H2

H1 H2 H3 H4 H5

Ping Hu (UIUC) MIGHTY LII 10 / 14

slide-34
SLIDE 34

Is the bound good?

ρ(G) =

  • H∈Hs

ρ(H)p(H, G) πQ(C4) ≤ max

H∈Hs ρ(H)

H2

H1 H2 H3 H4 H5

πQ(C4) ≤ max ρ(Hi) = ρ(H5) = 3/4

Ping Hu (UIUC) MIGHTY LII 10 / 14

slide-35
SLIDE 35

Is the bound good?

ρ(G) =

  • H∈Hs

ρ(H)p(H, G)

H1 H2 H3 H4 H5

πQ(C4) ≤ max ρ(Hi) = ρ(H5) = 3/4 If 0 ≤

i cHip(Hi, G), then

Ping Hu (UIUC) MIGHTY LII 10 / 14

slide-36
SLIDE 36

Is the bound good?

ρ(G) =

  • H∈Hs

ρ(H)p(H, G)

H1 H2 H3 H4 H5

πQ(C4) ≤ max ρ(Hi) = ρ(H5) = 3/4 If 0 ≤

i cHip(Hi, G), then

ρ(G) ≤

  • i

(ρ(Hi) + cHi) p(Hi, G) πQ(C4) ≤ max

i

(ρ(Hi) + cHi) cHi might be negative

Ping Hu (UIUC) MIGHTY LII 10 / 14

slide-37
SLIDE 37

Optimize cHi

Let M be a positive semidefinite 2-by-2 matrix. M = m11 m12 m21 m22

  • Ping Hu (UIUC)

MIGHTY LII 11 / 14

slide-38
SLIDE 38

Optimize cHi

Let M be a positive semidefinite 2-by-2 matrix. M = m11 m12 m21 m22

  • ρ(H1) + cH1

= 0 + m11 ρ(H2) + cH2 = 1/4 + m11/2 + m12/2 ρ(H3) + cH3 = 1/2 + m12 ρ(H4) + cH4 = 1/2 + m11/4 + m12/2 + m22/4 ρ(H5) + cH5 = 3/4 + m12/2 + m22/2 πQ(C4) ≤ max

i

(ρ(Hi) + cHi)

Ping Hu (UIUC) MIGHTY LII 11 / 14

slide-39
SLIDE 39

Solution

Take M =

  • 2/3

−1/3 −1/3 1/6

  • ,

then max

i

(ρ(Hi) + cHi) = 2/3

Ping Hu (UIUC) MIGHTY LII 12 / 14

slide-40
SLIDE 40

Results

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C4) ≤ 0.6068.

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C6) ≤ 0.3755. By using H3.

Ping Hu (UIUC) MIGHTY LII 13 / 14

slide-41
SLIDE 41

Results

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C4) ≤ 0.6068.

Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+])

πQ(C6) ≤ 0.3755. By using H3. Almost surely can be improved by waiting.

Ping Hu (UIUC) MIGHTY LII 13 / 14

slide-42
SLIDE 42

Thank you for your attention!

Ping Hu (UIUC) MIGHTY LII 14 / 14