lagrangians of hypergraphs - 上海交通大学数学系november 09, 2013 lagrangians of hypergraphs...

Lagrangians of Hypergraphs


Peng, Yuejian

Hunan University

November 09, 2013


Outline

1 Applications of Lagrangians in Turán type problem

2 Extension of Motzkin-Straus Theorem to some non-uniformhypergraphs

3 Some partial results to Frankl-Füredi conjecture


Applications of Lagrangians in Turán type problem

Outline






Lagrangian of an r-uniform graph

Lagrangian of an r-uniform graphG: 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) =12(1−

1t )

Remark

λ(G) ≥ |E|nr

for an r-uniform graph G with n vertices and edge set E.



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 thenλ(G) = λ(Kt) =

12(1−

1t ).



Turán type problem

Turán type problem

Question: For an r-uniform graph F and integer n, what is themaximum number of edges an r-uniform graph with n verticescan have without containing any member of F as a subgraph?

This number is denoted by ex(n, F ).



Turán density

Turán density

The extremal density (Turán density) of an r-uniform graphF is defined to be

π(F ) = limn→∞

ex(n, F )(nr

) .Remark.

An argument of Katona, Nemetz, Simonovits implies that such alimit exists.



Turán density

Theorem 2 (Turán, Mat. Fiz. Lapok 48 (1941))

π(Kt) = 1−1

t− 1.

Proof. Note that the complete (t− 1)-partite graph with nvertices whose partition sets differ in size by at most 1 does notcontain Kt. So π(Kt) ≥ 1− 1t−1 .

Let � > 0. If d(G) ≥ 1− 1t− 1

+ �, then

λ(G) ≥(1− 1t−1 + �)

(n2

)n2

≥ 1− 1t− 1

when n ≥ n(�).By Theorem 1, G must contain a kt.



Turán density

Theorem (Erdős-Stone-Simonovits, 1966)

Let F be a graph with at least 1 edge. Then

π(F ) = 1− 1χ(F )− 1

,

where χ(F ) is the chromatic number of F .

It can be proved by using the connection between Lagrangiansand Turán density of graphs.



Turán density

Applications in Spectral graph theory can be found in

H.S. Wilf, Spectral bounds for the clique and independencenumber of graphs, J. Combin. Theory Ser. B 40 (1986), 113-117.



Turán density

Applications in determining hypergraph Turan densities can befound in

1. A.F. Sidorenko, Solution of a problem of Bollobas on4-graphs, Mat. Zametki 41 (1987), 433-455.

2. P. Frankl and V. Rödl, Some Ramsey-Turán type results forhypergraphs, Combinatorica 8 (1989), 323-332.

3. P. Frankl and Z. Füredi, Extremal problems whosesolutions are the blow-ups of the small Witt-designs, Journal ofCombinatorial 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.



Turán density

Applications in finding hypergraph non-jumping numbers canbe found in

1. P. Frankl and V. Rödl, Hypergraphs do not jump,Combinatorica 4 (1984), 149-159.

2. P. Frankl, Y. Peng, V. Rödl and J. Talbot, A note on thejumping constant conjecture of Erdös, Journal of CombinatorialTheory 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 findnon-jumping numbers I, Discrete Mathematics 307 (2007),1754-1766.

5. Y. Peng, Using Lagrangians of hypergrpahs to findnon-jumping numbers II, Annals of Combinatorics 12 (2008), no.3, 307-324.

6. Y. Peng and C. Zhao, Generating non-jumping numbersrecursively, Discrete Applied Mathematics 156 (2008), no. 10,1856-1864.


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

Outline






Extension of Motzkin-Straus Theorem to

some non-uniform hypergraphs

A hypergraph H is a pair (V,E) with the vertex set V andedge set E ⊆ 2V .

The set R(H) = {|F | : F ∈ E} is called the set of its edgetypes.

For any k ∈ R(H), the level hypergraph Hk is thehypergraph consisting of all k-edges of H.

For a positive integer n and a subset R ⊂ [n], the completehypergraph KRn is a hypergraph on n vertices with edge set⋃i∈R([n]i

).



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 nvertices and R(G) ⊂ R(H) }.

π(H) = limn→∞

πn(H).

This concept is motivated by recent work on extremal posetproblems.



For an hypergraph HRn with R(H) = R, edge set E(H) and avector ~x = (x1, . . . , xn) ∈ Rn, define

λ(HRn , ~x) =∑j∈R

(j!∑

i1i2···ij∈Hjxi1xi2 . . . xij ).

Let S = {~x = (x1, x2, . . . , xn) :∑n

i=1 xi = 1, xi ≥ 0 for i =1, 2, . . . , n}. The Lagrangian of HRn , denoted by λ(HRn ), is definedas

λ(HRn ) = max{λ(HR, ~x) : ~x ∈ S}.



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− 1t .



Sketch of the proof.

Clearly, λ(H) ≥ λ(K{1,2}t ) = 2− 1t .Now show that λ(H) ≤ λ(K{1,2}t ) = 2− 1t . Let

~x = (x1, x2, . . . , xn) be an optimal weighting of H with k positiveweights such that the number of positive weights is minimized.Without loss of generalnality, we may assume thatx1 ≥ 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.



Throrem 4

For any hypergraph H = (V,E) with R(H) = {1, 2} and H2 isnot bipartite, π(H) = 2− 1

χ(H2)−1 .

This result is also proved by Johnston and Lu.



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 alle ∈ E(F ). We say that G is F − hom− free if there is nohomomorphism from F to G.

A hypergraph G is dense if every proper subgraph G′ satisfiesλ(G′) < λ(G).

Remark 1 A {1, 2}-graph G is dense if and only if G isK{1,2}t (t ≥ 2).

Lemma 3 π(F ) is the supremum of λ(G) over all denseF -hom-free G.

Therefore

π(H) = λ(K{1,2}t−1 ) = 2−

1

t− 1= 2− 1

χ(H2)− 1.



Theorem 5 (Gu-Li-Peng-Shi, submitted)

Let H be a {1, r2, · · · , rk}-hypergraph. If both the order of itsmaximum complete {1, r2, · · · , rk}-subgraph and the order of itsmaximum 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 maximumcomplete {1, 2, 3}-subgraph and the order of its maximumcomplete {1}-subgraph are t, where t ≥ 8, then

λ(H) = λ(Kt{1,2,3}

)= 1 +

t− 1t

+(t− 1)(t− 2)

t2.



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 maximumcomplete 3-graph of order s, where s ≥ t, and the number of edgesin H3 satisfies

(s3

)≤ e(H3) ≤

(s3

)+(t−12

), then,

λ′(H) = λ′(Kt{1,3}

)= 1 +

(t− 1)(t− 2)t2

.



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 notachieve their Lagrangian on any proper subhypergraph.


Some partial results to Frankl-Füredi conjecture

Outline






Frankl-Füredi Conjecture

In most applications, we need an upper bound for theLagrangian of a hypergraph.

Frankl and Füredi asked the followingQuestion. Given r ≥ 3 and m ∈ N how large can the Lagrangianof an r-graph with m edges be?



For distinct A,B ∈ N (r) we say that A is less than B in thecolex ordering if max(A4B) ∈ 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(tr

)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).



Conjecture 1 (Frankl-Füredi,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. FFConjecture is also true for r = 3 and m =

(l3

)− 1 or m =

(l3

)− 2.



Although, the obvious generalization of Motzkin and Straus’result to hypergraphs is false, we attempt to explore therelationship between the Lagrangian of a hypergraph and the sizeof its maximum cliques for hypergraphs when the number of edgesis in certain range.



Conjecture 2

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

)≤ m ≤

(l−1r

)+(l−2r−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−1r

)+(l−2r−1)

is the best possible.

When m =(l−1r

)+(l−2r−1)

+ 1, let ~x = (x1, . . . , xl), where

x1 = x2 = · · · = xl−2 = 1l−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 Gcontain no clique of size l − 1, where

(l−1r

)≤ m ≤

(l−1r

)+(l−2r−1).

Then λ(G) < λ(K(r)l−1).



Lemma 4 (Talbot, 2002)

For integers m and t satisfying(t−1r

)≤ m ≤

(t−1r

)+(t−2r−1), we

have λ(Cr,m) = λ(K(r)t−1]).

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

)≤ m ≤

(t−1r

)+(t−2r−1).



Theorem 8 (Peng-Zhao, Graphs and Combinatorics, 2013)

Let m and t be positive integers satisfying(t−13

)≤ m ≤

(t−13

)+(t−22

). 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 CombinatorialOptimization, accepted)

(a) Let m and t be positive integers satisfying(t−1r

)≤ m ≤

(t−1r

)+(t−2r−1)− (2r−3 − 1)(

(t−2r−2)− 1). Let G be an

r-graph on t vertices with m edges and contain a clique of ordert− 1. Then λ(G) = λ([t− 1](r)).(b) Let m and t be positive integers satisfying(t−13

)≤ m ≤

(t−13

)+(t−22

)− (t− 2). Let G be a 3-graph with m

edges and without containing a clique of order t− 1. Thenλ(G) < λ([t− 1](3)).

This result implies Talbot’s result.



Theorem 10 (Tang-Peng-Zhang-Zhao, Discrete AppliedMathematics, accepted)

Frankl-Furedi Conjecture holds for r ≥ 3 when(tr

)− 4 ≤ m ≤

(tr

).

Theorem 11 (Tang-Peng-Zhang-Zhao, manuscript)

Let m and t be integers satisfying(t−13

)≤ m ≤

(t−13

)+(t−22

)− 12(t− 1). Let G be a 3-graph with m

edges and G does not contain a clique order of t− 1, thenλ(G) < λ([t− 1](3)).

This result implies that Frankl-Furedi Conjecture holds forr = 3 and

(t−13

)≤ m ≤

(t−13

)+(t−22

)− 12(t− 1).



Definition

An r-graph G = (V,E) on the vertex set [n] is left-compressed ifj1j2 · · · jr ∈ E implies i1i2 · · · ir ∈ E provided ip ≤ jp for everyp, 1 ≤ p ≤ r.

Theorem 12 (Peng-Zhu-Zheng-Zhao, submitted)

When r = 3, to verify Conjecture 3, it is sufficient to check for allleft-compressed 3-uniform graphs on t vertices withm =

(t−13

)+(t−22

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



Theorem 13 (Talbot, CPC, 2002)

Let m, t and a satisfy −(t− 2) ≤ a ≤ (t− 5) andm =

(t−13

)+(t−22

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

3-graph with m edges. Then G and C3,m differ in at most2(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 edgessatisfying |E(G)∆E(C3,m)| ≤ 6. Then λ(G) ≤ λ(C3,m).

Corollary

FF conjecture is true for r = 3 and(t3

)− 6 ≤ m ≤

(t3

).



Theorem 15 (Sun-Peng-Tang, submitted)

Let G = (V,E) be a left-compressed 3-graph on the vertex set [t]with

(t−13

)≤ m ≤

(t−13

)+(t−22

)edges and not containing a clique

of order t− 1. If |E(t−1)t| ≤ 7, then λ(G) < λ([t− 1](3)).



Question: Let F be an r-uniform graph. What is the maximumLagrangian of an r-uniform graph can have without containing Fas a subgraph?

Denote this number by L(F ).

Remark

π(F ) ≤ r!L(F ).

Let � > 0. If d(G) ≥ r!L(F ) + �, then

λ(G) ≥(r!L(F ) + �)

(nr

)nr

≥ r!L(F )

when n ≥ n(�). So G must contain F as a subgraph.

lagrangians of hypergraphs - 上海交通大学数学系november 09, 2013 lagrangians of hypergraphs...

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