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 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) MIGHTY LII 1 / 14

Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 Ping Hu (UIUC) MIGHTY LII 2 / 14

Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 e ( G ) := | E ( G ) | Ping Hu (UIUC) MIGHTY LII 2 / 14

Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 e ( G ) := | E ( G ) | ex Q ( n , F ) := the maximum number of edges of a F -free subgraph of Q n Ping Hu (UIUC) MIGHTY LII 2 / 14

Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 e ( G ) := | E ( G ) | ex Q ( n , F ) := the maximum number of edges of a F -free subgraph of Q n ex Q ( n , F ) π Q ( F ) = lim e ( Q n ) n →∞ Ping Hu (UIUC) MIGHTY LII 2 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 Ping Hu (UIUC) MIGHTY LII 3 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 Q 7 Q 7 π Q ( C 4 ) ≥ 1 / 2 Ping Hu (UIUC) MIGHTY LII 3 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 Q 7 Q 7 π Q ( C 4 ) ≥ 1 / 2 Ping Hu (UIUC) MIGHTY LII 3 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 Theorem (Chung [1992], Brouwer–Dejter–Thomassen [1993]) π Q ( C 6 ) ≥ 1 / 4 Ping Hu (UIUC) MIGHTY LII 3 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 Theorem (Chung [1992], Brouwer–Dejter–Thomassen [1993]) π Q ( C 6 ) ≥ 1 / 4 Theorem (Conder [1993]) π Q ( C 6 ) ≥ 1 / 3 Ping Hu (UIUC) MIGHTY LII 3 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 . Theorem (Chung [1992]) π Q ( n , C 2 t ) = 0 for even t ≥ 4 . uredi–¨ Theorem (F¨ Ozkahya [2009]) π Q ( C 2 t ) = 0 for odd t ≥ 7 . Ping Hu (UIUC) MIGHTY LII 4 / 14

π Q ( C 2 t ) Conjecture (Erd˝ os [1984]) π Q ( C 4 ) = 1 / 2 , π Q ( C 2 t ) = 0 for t > 2 . Theorem (Chung [1992]) π Q ( n , C 2 t ) = 0 for even t ≥ 4 . uredi–¨ Theorem (F¨ Ozkahya [2009]) π Q ( C 2 t ) = 0 for odd t ≥ 7 . if π Q ( C 10 ) = 0 is still open. Ping Hu (UIUC) MIGHTY LII 4 / 14

π Q ( C 4 ) Theorem (Brass–Harborth–Nienborg [1995]) ex Q ( n , C 4 ) ≥ 1 1 2 (1 + √ n ) e ( Q n ) (valid when n is a power of 4 ) Ping Hu (UIUC) MIGHTY LII 5 / 14

π Q ( C 4 ) Theorem (Brass–Harborth–Nienborg [1995]) ex Q ( n , C 4 ) ≥ 1 1 2 (1 + √ n ) e ( Q n ) (valid when n is a power of 4 ) Theorem (Chung [1992]) π Q ( C 4 ) ≤ 0 . 62284 . Ping Hu (UIUC) MIGHTY LII 5 / 14

π Q ( C 4 ) Theorem (Brass–Harborth–Nienborg [1995]) ex Q ( n , C 4 ) ≥ 1 1 2 (1 + √ n ) e ( Q n ) (valid when n is a power of 4 ) Theorem (Chung [1992]) π Q ( C 4 ) ≤ 0 . 62284 . Theorem (Thomason–Wagner [2009]) π Q ( C 4 ) ≤ 0 . 62256 . Ping Hu (UIUC) MIGHTY LII 5 / 14

π Q ( C 4 ) Theorem (Brass–Harborth–Nienborg [1995]) ex Q ( n , C 4 ) ≥ 1 1 2 (1 + √ n ) e ( Q n ) (valid when n is a power of 4 ) Theorem (Chung [1992]) π Q ( C 4 ) ≤ 0 . 62284 . Theorem (Thomason–Wagner [2009]) π Q ( C 4 ) ≤ 0 . 62083 . Ping Hu (UIUC) MIGHTY LII 5 / 14

π Q ( C 4 ) Theorem (Brass–Harborth–Nienborg [1995]) ex Q ( n , C 4 ) ≥ 1 1 2 (1 + √ n ) e ( Q n ) (valid when n is a power of 4 ) Theorem (Chung [1992]) π Q ( C 4 ) ≤ 0 . 62284 . Theorem (Thomason–Wagner [2009]) π Q ( C 4 ) ≤ 0 . 62083 . Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+]) π Q ( C 4 ) ≤ 0 . 6068 . Ping Hu (UIUC) MIGHTY LII 5 / 14

π Q ( n , C 6 ) Theorem (Conder [1993]) π Q ( C 6 ) ≥ 1 / 3 . Ping Hu (UIUC) MIGHTY LII 6 / 14

π Q ( n , C 6 ) Theorem (Conder [1993]) π Q ( C 6 ) ≥ 1 / 3 . Theorem (Chung [1992]) √ π Q ( C 6 ) ≤ 2 − 1 ≈ 0 . 41421 . Ping Hu (UIUC) MIGHTY LII 6 / 14

π Q ( n , C 6 ) Theorem (Conder [1993]) π Q ( C 6 ) ≥ 1 / 3 . Theorem (Chung [1992]) √ π Q ( C 6 ) ≤ 2 − 1 ≈ 0 . 41421 . Theorem (Balogh–Hu–Lidick´ y–Liu, ind. Baber [2012+]) π Q ( C 6 ) ≤ 0 . 3755 . Ping Hu (UIUC) MIGHTY LII 6 / 14

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 G ∈G , | V ( G ) |→∞ p ( H , G ) . lim Ping Hu (UIUC) MIGHTY LII 7 / 14

Results using Flag Algebras Ping Hu (UIUC) MIGHTY LII 8 / 14

Results using Flag Algebras Theorem (Hladk´ y–Kr´ al’–Norine [2009]) Every n-vertex digraph with minimum outdegree at least 0 . 3465 n contains a triangle. Ping Hu (UIUC) MIGHTY LII 8 / 14

Results using Flag Algebras Theorem (Hladk´ y–Kr´ al’–Norine [2009]) Every n-vertex digraph with minimum outdegree at least 0 . 3465 n contains a triangle. Theorem (Hatami–Hladk´ y–Kr´ al’–Norine–Razborov [2011], Grzesik [2011]) The number of C 5 s in a triangle-free graph of order n is at most ( n / 5) 5 . Ping Hu (UIUC) MIGHTY LII 8 / 14

Results using Flag Algebras Theorem (Hladk´ y–Kr´ al’–Norine [2009]) Every n-vertex digraph with minimum outdegree at least 0 . 3465 n contains a triangle. Theorem (Hatami–Hladk´ y–Kr´ al’–Norine–Razborov [2011], Grzesik [2011]) The number of C 5 s in a triangle-free graph of order n is at most ( n / 5) 5 . Theorem (Falgas-Ravry–Vaughan [2011]) π ( K − 4 , C 5 , F 3 , 2 ) = 12 / 49 , π ( K − 4 , F 3 , 2 ) = 5 / 18 . F 3 , 2 : { 123 , 145 , 245 , 345 } , C 5 : { 123 , 234 , 345 , 451 , 512 } . Ping Hu (UIUC) MIGHTY LII 8 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Ping Hu (UIUC) MIGHTY LII 9 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Bound infinite problem by a finite piece. Ping Hu (UIUC) MIGHTY LII 9 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Bound infinite problem by a finite piece. Definition H n : the family of spanning subgraphs of Q n not containing C 4 . Let H ∈ H s , G ∈ H n , s < n , p ( H , G ) is the probability that a random s -hypercube vertex set in G induces H . ρ ( G ) = e ( G ) / e ( Q n ) . Ping Hu (UIUC) MIGHTY LII 9 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Bound infinite problem by a finite piece. Definition H n : the family of spanning subgraphs of Q n not containing C 4 . Let H ∈ H s , G ∈ H n , s < n , p ( H , G ) is the probability that a random s -hypercube vertex set in G induces H . ρ ( G ) = e ( G ) / e ( Q n ) . � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s Ping Hu (UIUC) MIGHTY LII 9 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Bound infinite problem by a finite piece. Definition H n : the family of spanning subgraphs of Q n not containing C 4 . Let H ∈ H s , G ∈ H n , s < n , p ( H , G ) is the probability that a random s -hypercube vertex set in G induces H . ρ ( G ) = e ( G ) / e ( Q n ) . � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s ρ ( G ) ≤ max H ∈H s ρ ( H ) Ping Hu (UIUC) MIGHTY LII 9 / 14

Proof by an Example Example π Q ( C 4 ) ≤ 2 / 3 Bound infinite problem by a finite piece. Definition H n : the family of spanning subgraphs of Q n not containing C 4 . Let H ∈ H s , G ∈ H n , s < n , p ( H , G ) is the probability that a random s -hypercube vertex set in G induces H . ρ ( G ) = e ( G ) / e ( Q n ) . � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s π Q ( C 4 ) ≤ max H ∈H s ρ ( H ) Ping Hu (UIUC) MIGHTY LII 9 / 14

Is the bound good? � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s π Q ( C 4 ) ≤ max H ∈H s ρ ( H ) Ping Hu (UIUC) MIGHTY LII 10 / 14

Is the bound good? � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s π Q ( C 4 ) ≤ max H ∈H s ρ ( H ) H 2 H 1 H 2 H 3 H 4 H 5 Ping Hu (UIUC) MIGHTY LII 10 / 14

Is the bound good? � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s π Q ( C 4 ) ≤ max H ∈H s ρ ( H ) H 2 H 1 H 2 H 3 H 4 H 5 π Q ( C 4 ) ≤ max ρ ( H i ) = ρ ( H 5 ) = 3 / 4 Ping Hu (UIUC) MIGHTY LII 10 / 14

Is the bound good? � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s H 1 H 2 H 3 H 4 H 5 π Q ( C 4 ) ≤ max ρ ( H i ) = ρ ( H 5 ) = 3 / 4 If 0 ≤ � i c H i p ( H i , G ), then Ping Hu (UIUC) MIGHTY LII 10 / 14

Is the bound good? � ρ ( G ) = ρ ( H ) p ( H , G ) H ∈H s H 1 H 2 H 3 H 4 H 5 π Q ( C 4 ) ≤ max ρ ( H i ) = ρ ( H 5 ) = 3 / 4 If 0 ≤ � i c H i p ( H i , G ), then � ρ ( G ) ≤ ( ρ ( H i ) + c H i ) p ( H i , G ) i π Q ( C 4 ) ≤ max ( ρ ( H i ) + c H i ) i c H i might be negative Ping Hu (UIUC) MIGHTY LII 10 / 14

Optimize c H i Let M be a positive semidefinite 2-by-2 matrix. � m 11 � m 12 M = m 21 m 22 Ping Hu (UIUC) MIGHTY LII 11 / 14