Lagrangians of Hypergraphs Peng, Yuejian Hunan University November - PowerPoint PPT Presentation

Lagrangians of Hypergraphs Lagrangians of Hypergraphs Peng, Yuejian Hunan University November 09, 2013

Lagrangians of Hypergraphs Outline Applications of Lagrangians in Tur´ an type problem 1 Extension of Motzkin-Straus Theorem to some non-uniform 2 hypergraphs Some partial results to Frankl-F¨ uredi conjecture 3

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Outline Applications of Lagrangians in Tur´ an type problem 1 Extension of Motzkin-Straus Theorem to some non-uniform 2 hypergraphs Some partial results to Frankl-F¨ uredi conjecture 3

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Lagrangian of an r -uniform graph Lagrangian of an r -uniform graph G : an r -uniform graph with vertex set { 1 , 2 , . . . , n } and edge set E . x = ( x 1 , . . . , x n ) ∈ R n , where � n � i =1 x i = 1 , x i ≥ 0 . � λ ( G, � x ) = x i 1 · · · x i r . { i 1 ,...,i r }∈ E λ ( G ) = max { λ ( G, � x ) } . Example: λ ( K t ) = 1 2 (1 − 1 t ) Remark λ ( G ) ≥ | E | n r for an r -uniform graph G with n vertices and edge set E .

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Lagrangian of an r -uniform graph Theorem 1 (Motzkin and Straus, Canad. J. Math 17 (1965)) If G is a 2-graph in which a largest clique has order t then λ ( G ) = λ ( K t ) = 1 2 (1 − 1 t ) .

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an type problem Tur´ an type problem Question: For an r -uniform graph F and integer n , what is the maximum number of edges an r -uniform graph with n vertices can have without containing any member of F as a subgraph? This number is denoted by ex( n, F ) .

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Tur´ an density The extremal density (Tur´ an density) of an r -uniform graph F is defined to be ex( n, F ) π ( F ) = lim . � n � n →∞ r Remark. An argument of Katona, Nemetz, Simonovits implies that such a limit exists.

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Theorem 2 (Tur´ an, Mat. Fiz. Lapok 48 (1941)) 1 π ( K t ) = 1 − t − 1 . Proof. Note that the complete ( t − 1) -partite graph with n vertices whose partition sets differ in size by at most 1 does not 1 contain K t . So π ( K t ) ≥ 1 − t − 1 . 1 Let ǫ > 0 . If d ( G ) ≥ 1 − t − 1 + ǫ , then 1 � n � (1 − t − 1 + ǫ ) 1 2 λ ( G ) ≥ ≥ 1 − n 2 t − 1 when n ≥ n ( ǫ ) . By Theorem 1, G must contain a k t .

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Theorem (Erd˝ os-Stone-Simonovits, 1966) Let F be a graph with at least 1 edge. Then 1 π ( F ) = 1 − χ ( F ) − 1 , where χ ( F ) is the chromatic number of F . It can be proved by using the connection between Lagrangians and Tur´ an density of graphs.

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Applications in Spectral graph theory can be found in H.S. Wilf, Spectral bounds for the clique and independence number of graphs, J. Combin. Theory Ser. B 40 (1986), 113-117.

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Applications in determining hypergraph Turan densities can be found in 1. A.F. Sidorenko, Solution of a problem of Bollobas on 4-graphs, Mat. Zametki 41 (1987), 433-455. 2. P. Frankl and V. R¨ odl, Some Ramsey-Tur´ an type results for hypergraphs, Combinatorica 8 (1989), 323-332. 3. P. Frankl and Z. F¨ uredi, Extremal problems whose solutions are the blow-ups of the small Witt-designs, Journal of Combinatorial Theory (A) 52 (1989), 129-147. 4. D. Mubayi, A hypergraph extension of Turan’s theorem, J. Combin. Theory Ser. B 96 (2006), 122-134.

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density Applications in finding hypergraph non-jumping numbers can be found in 1. P. Frankl and V. R¨ odl, Hypergraphs do not jump, Combinatorica 4 (1984), 149-159. 2. P. Frankl, Y. Peng, V. R¨ odl and J. Talbot, A note on the jumping constant conjecture of Erd¨ os, Journal of Combinatorial Theory Ser. B. 97 (2007), 204-216. 3. Y. Peng, Non-jumping numbers for 4-uniform hypergraphs, Graphs and Combinatorics 23 (2007), 97-110. 4. Y. Peng, Using Lagrangians of hypergrpahs to find non-jumping numbers I, Discrete Mathematics 307 (2007), 1754-1766. 5. Y. Peng, Using Lagrangians of hypergrpahs to find non-jumping numbers II, Annals of Combinatorics 12 (2008), no. 3, 307-324. 6. Y. Peng and C. Zhao, Generating non-jumping numbers recursively, Discrete Applied Mathematics 156 (2008), no. 10, 1856-1864.

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs Outline Applications of Lagrangians in Tur´ an type problem 1 Extension of Motzkin-Straus Theorem to some non-uniform 2 hypergraphs Some partial results to Frankl-F¨ uredi conjecture 3

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs A hypergraph H is a pair ( V, E ) with the vertex set V and edge set E ⊆ 2 V . The set R ( H ) = {| F | : F ∈ E } is called the set of its edge types . For any k ∈ R ( H ) , the level hypergraph H k is the hypergraph consisting of all k -edges of H . For a positive integer n and a subset R ⊂ [ n ] , the complete hypergraph K R n is a hypergraph on n vertices with edge set � [ n ] � � . i ∈ R i

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs For a non-uniform hypergraph G on n vertices, define 1 � h n ( G ) = � . � n | F | F ∈ E ( G ) π n ( H ) = max { h n ( G ) : G is a H -free hypergraph with n vertices and R ( G ) ⊂ R ( H ) } . π ( H ) = lim n →∞ π n ( H ) . This concept is motivated by recent work on extremal poset problems.

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs For an hypergraph H R n with R ( H ) = R , edge set E ( H ) and a x = ( x 1 , . . . , x n ) ∈ R n , define vector � � � λ ( H R n , � x ) = ( j ! x i 1 x i 2 . . . x i j ) . j ∈ R i 1 i 2 ··· i j ∈ H j x = ( x 1 , x 2 , . . . , x n ) : � n Let S = { � i =1 x i = 1 , x i ≥ 0 for i = 1 , 2 , . . . , n } . The Lagrangian of H R n , denoted by λ ( H R n ) , is defined as λ ( H R n ) = max { λ ( H R , � x ) : � x ∈ S } .

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs If R ( H ) = { 1 , 2 } , then H is called a { 1 , 2 } -graph. Theorem 3 (Peng-Peng-Tang-Zhao, submitted) If H is a { 1 , 2 } -graph and the order of its maximum complete { 1 , 2 } -subgraph is t ( t ≥ 2) , then λ ( H ) = λ ( K { 1 , 2 } ) = 2 − 1 t . t

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs Sketch of the proof. Clearly, λ ( H ) ≥ λ ( K { 1 , 2 } ) = 2 − 1 t . t Now show that λ ( H ) ≤ λ ( K { 1 , 2 } ) = 2 − 1 t . Let t � x = ( x 1 , x 2 , . . . , x n ) be an optimal weighting of H with k positive weights such that the number of positive weights is minimized. Without loss of generalnality, we may assume that x 1 ≥ x 2 ≥ . . . ≥ x k > x k +1 = x k +2 = . . . x n = 0 . Lemma 1 ∂λ ( H,� x ) = ∂λ ( H,� x ) = . . . = ∂λ ( H,� x ) . ∂x 1 ∂x 2 ∂x k Lemma 2 ∀ 1 ≤ i < j ≤ k, ij ∈ H 2 . Claim 1 ∀ 1 ≤ i ≤ k , if i ∈ H but j / ∈ H , then x i − x j = 0 . 5 . Claim 2 Either the theorem holds or i ∈ H 1 for all 1 ≤ i ≤ k . So K { 1 , 2 } is a subgraph of H . Since t is the order of the k maximum complete { 1 , 2 } -graph of H , then k ≤ t . We have ) = 2 − 1 k ≤ 2 − 1 x ) = λ ( K { 1 , 2 } λ ( H, � t . k

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs Throrem 4 For any hypergraph H = ( V, E ) with R ( H ) = { 1 , 2 } and H 2 is 1 not bipartite, π ( H ) = 2 − χ ( H 2 ) − 1 . This result is also proved by Johnston and Lu.

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs Sketch of the proof. f : V ( F ) → V ( G ) is called a homomorphism form hypegraph F to hypergraph G if it preserves edges, i.e. f ( e ) ∈ E ( G ) for all e ∈ E ( F ) . We say that G is F − hom − free if there is no homomorphism from F to G . A hypergraph G is dense if every proper subgraph G ′ satisfies λ ( G ′ ) < λ ( G ) . Remark 1 A { 1 , 2 } -graph G is dense if and only if G is K { 1 , 2 } ( t ≥ 2) . t Lemma 3 π ( F ) is the supremum of λ ( G ) over all dense F -hom-free G . Therefore 1 1 π ( H ) = λ ( K { 1 , 2 } t − 1 ) = 2 − t − 1 = 2 − χ ( H 2 ) − 1 .