Lagrangians of Hypergraphs
Lagrangians of Hypergraphs
Peng, Yuejian
Hunan University
November 09, 2013
Lagrangians of Hypergraphs
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
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán 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 thenλ(G) = λ(Kt) =
12(1−
1t ).
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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 ).
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Applications of Lagrangians in Turán type problem
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.
Lagrangians of Hypergraphs
Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs
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
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 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
).
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 nvertices and R(G) ⊂ R(H) }.
π(H) = limn→∞
πn(H).
This concept is motivated by recent work on extremal posetproblems.
Lagrangians of Hypergraphs
Extension of Motzkin-Straus Theorem to some non-uniform hypergraphs
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}.
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− 1t .
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− 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.
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 isnot 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 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.
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 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.
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 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
.
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 notachieve their Lagrangian on any proper subhypergraph.
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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?
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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.
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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.
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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.
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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.
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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
).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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)).
Lagrangians of Hypergraphs
Some partial results to Frankl-Füredi conjecture
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.