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Lagrangians of Hypergraphs

Lagrangians of Hypergraphs

Peng, Yuejian

Hunan University

November 09, 2013

Lagrangians of Hypergraphs

Outline

1

Applications of Lagrangians in Tur´ an type problem

2

Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

3

Some partial results to Frankl-F¨ uredi conjecture

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem

Outline

1

Applications of Lagrangians in Tur´ an type problem

2

Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

3

Some partial results to Frankl-F¨ uredi conjecture

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 = (x1, . . . , xn) ∈ Rn, where n

i=1 xi = 1, xi ≥ 0.

λ(G, x) =

• {i1,...,ir}∈E

xi1 · · · xir. λ(G) = max{λ(G, x)}. Example: λ(Kt) = 1

2(1 − 1 t )

Remark λ(G) ≥ |E| nr 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(Kt) = 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 π(F) = lim

n→∞

ex(n, F) 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)) π(Kt) = 1 − 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 contain Kt. So π(Kt) ≥ 1 −

1 t−1.

Let ǫ > 0. If d(G) ≥ 1 − 1 t − 1 + ǫ, then λ(G) ≥ (1 −

1 t−1 + ǫ)

n

2

• n2

≥ 1 − 1 t − 1 when n ≥ n(ǫ). By Theorem 1, G must contain a kt.

Lagrangians of Hypergraphs Applications of Lagrangians in Tur´ an type problem Tur´ an density

Theorem (Erd˝

• s-Stone-Simonovits, 1966)

Let F be a graph with at least 1 edge. Then π(F) = 1 − 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¨
• dl, 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¨
• dl, Hypergraphs do not jump,

Combinatorica 4 (1984), 149-159.

• 2. P. Frankl, Y. Peng, V. R¨
• dl and J. Talbot, A note on the

jumping constant conjecture of Erd¨

• s, 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

1

Applications of Lagrangians in Tur´ an type problem

2

Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

3

Some partial results to Frankl-F¨ uredi conjecture

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 ⊆ 2V . The set R(H) = {|F| : F ∈ E} is called the set of its edge types. For any k ∈ R(H), the level hypergraph Hk is the hypergraph consisting of all k-edges of H. For a positive integer n and a subset R ⊂ [n], the complete hypergraph KR

n is a hypergraph on n vertices with edge set

• i∈R

[n]

i

• .

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

For a non-uniform hypergraph G on n vertices, define hn(G) =

• F∈E(G)

1 n

|F|

. πn(H) = max{hn(G) : G is a H-free hypergraph with n vertices and R(G) ⊂ R(H) }. π(H) = lim

n→∞ πn(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 HR

n with R(H) = R, edge set E(H) and a

vector x = (x1, . . . , xn) ∈ Rn, define λ(HR

n ,

x) =

• j∈R

(j!

• i1i2···ij∈Hj

xi1xi2 . . . xij). Let S = { x = (x1, x2, . . . , xn) : n

i=1 xi = 1, xi ≥ 0 for i =

1, 2, . . . , n}. The Lagrangian of HR

n , denoted by λ(HR n ), is defined

as λ(HR

n ) = max{λ(HR,

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}

t

) = 2 − 1

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}

t

) = 2 − 1

t .

Now show that λ(H) ≤ λ(K{1,2}

t

) = 2 − 1

t . Let

• x = (x1, x2, . . . , xn) 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 x1 ≥ x2 ≥ . . . ≥ xk > xk+1 = xk+2 = . . . xn = 0. Lemma 1 ∂λ(H,

x) ∂x1

= ∂λ(H,

x) ∂x2

= . . . = ∂λ(H,

x) ∂xk

. Lemma 2 ∀1 ≤ i < j ≤ k, ij ∈ H2. Claim 1 ∀1 ≤ i ≤ k, if i ∈ H but j / ∈ H, then xi − xj = 0.5. Claim 2 Either the theorem holds or i ∈ H1 for all 1 ≤ i ≤ k. So K{1,2}

k

is a subgraph of H. Since t is the order of the maximum complete {1, 2}-graph of H, then k ≤ t. We have λ(H, x) = λ(K{1,2}

k

) = 2 − 1 k ≤ 2 − 1 t .

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 H2 is not bipartite, π(H) = 2 −

1 χ(H2)−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

(t ≥ 2). Lemma 3 π(F) is the supremum of λ(G) over all dense F-hom-free G. Therefore π(H) = λ(K{1,2}

t−1 ) = 2 −

1 t − 1 = 2 − 1 χ(H2) − 1.

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

Theorem 5 (Gu-Li-Peng-Shi, submitted) Let H be a {1, r2, · · · , rk}-hypergraph. If both the order of its maximum complete {1, r2, · · · , rk}-subgraph and the order of its maximum complete {1}-subgraph are t, where t ≥ f(r2, · · · , rk) for some function f(r2, · · · , rk), then λ(H) = λ

• Kt{1,r2,··· ,rk}

. Theorem 6 Let H be a {1, 2, 3}-graph. If both the order of its maximum complete {1, 2, 3}-subgraph and the order of its maximum complete {1}-subgraph are t, where t ≥ 8, then λ(H) = λ

• Kt{1,2,3}

= 1 + t − 1 t + (t − 1)(t − 2) t2 .

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

Theorem 7 Let H be a {1, 3}-graph. If the order of its maximum complete {1, 3}-subgraph is t, where t ≥ 5, H3 contains a maximum complete 3-graph of order s, where s ≥ t, and the number of edges in H3 satisfies s

3

• ≤ e(H3) ≤

s

3

• +

t−1

2

• , then,

λ′(H) = λ′ Kt{1,3} = 1 + (t − 1)(t − 2) t2 .

Lagrangians of Hypergraphs Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

Question: For an r-hypergraph G, does λ(G) = λ(K(r)

t

) always hold if K(r)

t

is a maximum clique of G? No! There are many examples of r-hypergraphs that do not achieve their Lagrangian on any proper subhypergraph.

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Outline

1

Applications of Lagrangians in Tur´ an type problem

2

Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs

3

Some partial results to Frankl-F¨ uredi conjecture

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Frankl-F¨ uredi Conjecture

In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and F¨ uredi asked the following

• Question. Given r ≥ 3 and m ∈ N how large can the Lagrangian
• f an r-graph with m edges be?

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

For distinct A, B ∈ N(r) we say that A is less than B in the colex ordering if max(A△B) ∈ B. In colex ordering, 123 < 124 < 134 < 234 < 125 < 135 < 235 < 145 < 245 < 345 < 126 < 136 < 236 < 146 < 246 < 346 < 156 < 256 < 356 < 456 < 127 < · · · . The first t

r

• r-tuples in the colex ordering of N(r) are the

edges of [t](r). Cr,m denotes the r-graph with m edges formed by taking the first m sets in the colex ordering of N(r).

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Conjecture 1 (Frankl-F¨ uredi,1989) For any r-graph G with m edges, λ(G) ≤ λ(Cr,m). FF conjecture is true when r = 2 by Motzkin-Straus’s result. Theorem 7 (Talbot, Combinatorics, Probability & Computing 11, 2002) Let m and l be integers satisfying l − 1 3

• ≤ m ≤

l − 1 3

• +

l − 2 2

• − (l − 1).

Then FF Conjecture is true for r = 3 and this value of m. FF Conjecture is also true for r = 3 and m = l

3

• − 1 or m =

l

3

• − 2.

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Although, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false, we attempt to explore the relationship between the Lagrangian of a hypergraph and the size

• f its maximum cliques for hypergraphs when the number of edges

is in certain range.

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Conjecture 2 Let l, m and r ≥ 3 be positive integers satisfying l−1

r

• ≤ m ≤

l−1

r

• +

l−2

r−1

• . Let G be an r-graph with m edges

and G contain a clique of order l − 1. Then λ(G) = λ(K(r)

l−1).

The upper bound l−1

r

• +

l−2

r−1

• is the best possible.

When m = l−1

r

• +

l−2

r−1

• + 1, let

x = (x1, . . . , xl), where x1 = x2 = · · · = xl−2 =

1 l−1 and xl−1 = xl = 1 2(l−1).

Then λ(Cr,m) ≥ λ(Cr,m, x) > λ(K(r)

l−1).

Conjecture 3 Let G be an r-graph with m edges and l vertices, and let G contain no clique of size l − 1, where l−1

r

• ≤ m ≤

l−1

r

• +

l−2

r−1

• .

Then λ(G) < λ(K(r)

l−1).

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Lemma 4 (Talbot, 2002) For integers m and t satisfying t−1

r

• ≤ m ≤

t−1

r

• +

t−2

r−1

• , we

have λ(Cr,m) = λ(K(r)

t−1]).

Conjectures 2 and 3 imply that FF Conjecture is true when t−1

r

• ≤ m ≤

t−1

r

• +

t−2

r−1

• .

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Theorem 8 (Peng-Zhao, Graphs and Combinatorics, 2013) Let m and t be positive integers satisfying t−1

3

• ≤ m ≤

t−1

3

• +

t−2

2

• . Let G be a 3-graph with m edges and

G contain a clique of order t − 1. Then λ(G) = λ(K(3)

t−1).

Theorem 9 (Peng-Tang-Zhao, Journal of Combinatorial Optimization, accepted) (a) Let m and t be positive integers satisfying t−1

r

• ≤ m ≤

t−1

r

• +

t−2

r−1

• − (2r−3 − 1)(

t−2

r−2

• − 1). Let G be an

r-graph on t vertices with m edges and contain a clique of order t − 1. Then λ(G) = λ([t − 1](r)). (b) Let m and t be positive integers satisfying t−1

3

• ≤ m ≤

t−1

3

• +

t−2

2

• − (t − 2). Let G be a 3-graph with m

edges and without containing a clique of order t − 1. Then λ(G) < λ([t − 1](3)). This result implies Talbot’s result.

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Theorem 10 (Tang-Peng-Zhang-Zhao, Discrete Applied Mathematics, accepted) Frankl-Furedi Conjecture holds for r ≥ 3 when t

r

• − 4 ≤ m ≤

t

r

• .

Theorem 11 (Tang-Peng-Zhang-Zhao, manuscript) Let m and t be integers satisfying t−1

3

• ≤ m ≤

t−1

3

• +

t−2

2

• − 1

2(t − 1). Let G be a 3-graph with m

edges and G does not contain a clique order of t − 1, then λ(G) < λ([t − 1](3)). This result implies that Frankl-Furedi Conjecture holds for r = 3 and t−1

3

• ≤ m ≤

t−1

3

• +

t−2

2

• − 1

2(t − 1).

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Definition An r-graph G = (V, E) on the vertex set [n] is left-compressed if j1j2 · · · jr ∈ E implies i1i2 · · · ir ∈ E provided ip ≤ jp for every p, 1 ≤ p ≤ r. Theorem 12 (Peng-Zhu-Zheng-Zhao, submitted) When r = 3, to verify Conjecture 3, it is sufficient to check for all left-compressed 3-uniform graphs on t vertices with m = t−1

3

• +

t−2

2

• edges without containing a clique of order l − 1.

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Theorem 13 (Talbot, CPC, 2002) Let m, t and a satisfy −(t − 2) ≤ a ≤ (t − 5) and m = t−1

3

• +

t−2

2

• + a. Suppose G is a left-compressed extremal

3-graph with m edges. Then G and C3,m differ in at most 2(t − a − 2) edges, i.e., |E(G)∆E(C3,m)| ≤ 2(t − a − 2). Theorem 14 (Tang-Peng-Wang-Peng, submitted) Let m be any positive integer. Let G be a 3-graph with m edges satisfying |E(G)∆E(C3,m)| ≤ 6. Then λ(G) ≤ λ(C3,m). Corollary FF conjecture is true for r = 3 and t

3

• − 6 ≤ m ≤

t

3

• .

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Theorem 15 (Sun-Peng-Tang, submitted) Let G = (V, E) be a left-compressed 3-graph on the vertex set [t] with t−1

3

• ≤ m ≤

t−1

3

• +

t−2

2

• edges and not containing a clique
• f order t − 1. If |E(t−1)t| ≤ 7, then λ(G) < λ([t − 1](3)).

Lagrangians of Hypergraphs Some partial results to Frankl-F¨ uredi conjecture

Question: Let F be an r-uniform graph. What is the maximum Lagrangian of an r-uniform graph can have without containing F as a subgraph? Denote this number by L(F). Remark π(F) ≤ r!L(F). Let ǫ > 0. If d(G) ≥ r!L(F) + ǫ, then λ(G) ≥ (r!L(F) + ǫ) n

r

• nr

≥ r!L(F) when n ≥ n(ǫ). So G must contain F as a subgraph.