On Lagrangians of Hypergraphs Containing Dense Subgraphs
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On Lagrangians of Hypergraphs Containing Dense Subgraphs
Qingsong Tang · Yuejian Peng · Xiangde Zhang · Cheng Zhao
Received: date / Accepted: date
Abstract
Motzkin and Straus established a remarkable connection between the maximum clique and the La-grangian of a graph in 1965. This connection and its extensions were successfully employed in optimization toprovide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hy-pergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphswhen the number of edges of the hypergraph is in certain ranges. These results support a pair of conjecturesintroduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002).
Keywords
Cliques of hypergraphs · Colex ordering · Lagrangians of hypergraphs · Polynomial optimization
Mathematics Subject Classification (2010) · · · In 1941, Tur´an [1] provided an answer to the following question: What is the maximum number of edges in agraph with n vertices not containing a complete subgraph of order k, for a given k? This is the well-known Tur´antheorem. Later, in another classical paper, Motzkin and Straus [2] provided a new proof of Tur´an theorem basedon the continuous characterization of the clique number of a graph using Lagrangians of graphs.
Qingsong TangCollege of Sciences, Northeastern University, Shenyang, 110819, P.R.China.School of Mathematics, Jilin University, Changchun 130012, P.R. China.E-mail: t [email protected] PengSchool of Mathematics, Hunan University, Changsha 410082, P.R. China.This research is supported by National Natural Science Foundation of China (No. 11271116).E-mail: [email protected] ZhangCollege of Sciences, Northeastern University, Shenyang, 110819, P.R.ChinaE-mail: [email protected] Zhao(Corresponding author)Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809 USA.School of Mathematics, Jilin University, Changchun 130012, P.R. China.E-mail: [email protected] Qingsong Tang et al.
The Motzkin-Straus result basically says that the Lagrangian of a graph which is the maximum of a homoge-neous quadratic multilinear function (determined by the graph) over the standard simplex of the Euclidean planeis connected to the maximum clique number of this graph (the precise statement is given in Theorem 2.1). Thisresult provides a solution to the optimization problem for a class of homogeneous quadratic multilinear functionsover the standard simplex of an Euclidean plane. The Motzkin-Straus result and its extension were successfullyemployed in optimization to provide heuristics for the maximum clique problem [3–6]. It has been also gener-alized to vertex-weighted graphs [6] and edge-weighted graphs with applications to pattern recognition in imageanalysis [3–9] The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. For ex-ample, Sidorenko [10] and Frankl-Furedi [11] applied Lagrangians of hypergraphs in finding Tur´an densities ofhypergraphs. Frankl and R¨odl [12] applied it in disproving Erd¨os long standing jumping constant conjecture. Inmost applications, we need an upper bound for the Lagrangian of a hypergraph.An attempt to generalize the Motzkin-Straus theorem to hypergraphs is due to S´os and Straus[13]. Recently,in [14, 15] Rota Bul´o and Pelillo generalized the Motzkin and Straus’ result to r -graphs in some way using a con-tinuous characterization of maximal cliques other than Lagrangians of hypergraphs. The obvious generalizationof Motzkin and Straus’ result to hypergraphs is false. In fact, there are many examples of hypergraphs that donot achieve their Lagrangian on any proper subhypergraph. We attempt to explore the relationship between theLagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges isin certain ranges though the obvious generalization of Motzkin and Straus’ result to hypergraphs is false.The results presented in Sect. 3 and 4 in this paper provide substantial evidence for two conjectures in [16]and extend some known results in the literature [16, 17]. The main results provide solutions to the optimizationproblem of a class of homogeneous multilinear functions over the standard simplex of the Euclidean space. Themain results also give connections between a continuous optimization problem and the maximum clique problemof hypergraphs. Since practical problems such as computer vision and image analysis are related to the maximumclique problems, this type of results opens a door to such practical applications. The results in this paper can be ap-plied in estimating Lagrangians of some hypergraphs, for example, calculations involving estimating Lagrangiansof several hypergraphs in [11] can be much simplified when applying the results in this paper.The rest of the paper is organized as follows. In Sect. 2, we state a few definitions, problems, and preliminaryresults. In Sect. 3 and Sect. 4, we provide upper bounds on the Lagrangian of a hypergraph containing densesubgraphs when the number of edges of the hypergraph is in a certain range. Then, as an application, using themain result in Sect. 3, we extend a result in [17] in Sect. 5. In Sect. 6, we give the proofs of some lemmas.Conclusions are given in Section 7. For a set V and a positive integer r we denote by V ( r ) the family of all r -subsets of V . An r -uniform graph or r -graph G consists of a set V ( G ) of vertices and a set E ( G ) ⊆ V ( G ) ( r ) of edges. An edge e : = { a , a , . . ., a r } willbe simply denoted by a a . . . a r . An r -graph H is a subgraph of an r -graph G , denoted by H ⊆ G if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ) . Let K ( r ) t denote the complete r -graph on t vertices, that is the r -graph on t vertices containingall possible edges. A complete r -graph on t vertices is also called a clique with order t . Let N be the set of allpositive integers. For any integer n ∈ N , we denote the set { , , , . . . , n } by [ n ] . Let [ n ] ( r ) represent the complete r -uniform graph on the vertex set [ n ] . When r =
2, an r -uniform graph is a simple graph. When r ≥
3, an r -graphis often called a hypergraph.For an r -graph G = ( V , E ) and i ∈ V , let E i : = { A ∈ V ( r − ) : A ∪ { i } ∈ E } . For a pair of vertices i , j ∈ V , let E i j : = { B ∈ V ( r − ) : B ∪ { i , j } ∈ E } . Let E ci : = { A ∈ V ( r − ) : A ∪ { i } ∈ V ( r ) \ E } , E ci j : = { B ∈ V ( r − ) : B ∪ { i , j } ∈ V ( r ) \ E } , and E i \ j : = E i ∩ E cj . n Lagrangians of Hypergraphs Containing Dense Subgraphs 3 Definition 2.1
For an r -uniform graph G with the vertex set [ n ] , edge set E ( G ) , and a vector x : = ( x , . . ., x n ) ∈ R n , we associate a homogeneous polynomial in n variables, denoted by l ( G , x ) as follows: l ( G , x ) : = (cid:229) i i ··· i r ∈ E ( G ) x i x i . . . x i r . Let S : = { x : = ( x , x , . . ., x n ) : (cid:229) ni = x i = , x i ≥ i = , , . . ., n } . Let l ( G ) represent the maximum of theabove homogeneous multilinear polynomial of degree r over the standard simplex S . Precisely l ( G ) : = max { l ( G , x ) : x ∈ S } . The value x i is called the weight of the vertex i . A vector x : = ( x , x , . . ., x n ) ∈ R n is called a feasible weightingfor G if x ∈ S . A vector y ∈ S is called an optimal weighting for G if l ( G , y ) = l ( G ) . Remark 2.1
Since l ( G ) is the maximum of a polynomial function in n variables x , x , · · · , x n under the constraint (cid:229) ni = x i = l ( G ) , l ( G ) was calledthe Lagrangian of G in several papers [11, 12, 17, 18]. Throughout this paper, we also call l ( G ) the Lagrangianof G .The following fact is easily implied by Definition 2.1. Fact 2.1
Let G , G be r-uniform graphs and G ⊆ G . Then l ( G ) ≤ l ( G ) . In [2], Motzkin and Straus provided the following simple expression for the Lagrangian of a 2-graph.
Theorem 2.1 (See [2], Theorem 1)
If G is a 2-graph with n vertices in which a largest clique has order t then l ( G ) = l ( K ( ) t ) = ( − t ) . Furthermore, the vector x : = ( x , x , . . ., x n ) given by x i : = t if i is a vertex in a fixedmaximum clique and x i = otherwise is an optimal weighting. This result provides a solution to the optimazation problem of this type of homogeneous quadratic functionsover the standard simplex of an Euclidean plane. It is well-known that Lagrangians of hypergraphs have beenproved to be a useful tool in hypergraph extremal problems, for example, it has been applied in finding Tur´an den-sities of hypergraphs in [10, 11, 18]. In order to explore the relationship between the Lagrangian of a hypergraphand the order of its maximum cliques for hypergraphs when the number of edges is in certain ranges, the followingtwo conjectures are proposed in [17].
Conjecture 2.1 (See [16], Conjecture 1.3) Let m and t be positive integers satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) . Let G be an r -graph with m edges and contain a clique of order t −
1. Then l ( G ) = l ([ t − ] ( r ) ) . Conjecture 2.2 (See [16], Conjecture 1.4) Let m and t be positive integers satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) . Let G be an r -graph with m edges and contain no clique of order t −
1. Then l ( G ) < l ([ t − ] ( r ) ) .In [16], we proved that Conjecture 2.1 holds for r = Theorem 2.2 (See [16], Theorem 1.8)
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . LetG be a -graph with m edges and contain a clique of order t − . Then l ( G ) = l ([ t − ] ( ) ) . For distinct A , B ∈ N ( r ) we say that A is less than B in the colex ordering iff max ( A △ B ) ∈ B , where A △ B : = ( A \ B ) ∪ ( B \ A ) . For example we have 246 <
156 in N ( ) since max ( { , , }△{ , , } ) ∈ { , , } . Incolex ordering, 123 < < < < < < < < < < < < < < < < < < < < < · · · . Note that the first (cid:0) tr (cid:1) r -tuples in the colex ordering of N ( r ) arethe edges of [ t ] ( r ) .Let C r , m denote the r -graph with m edges formed by taking the first m sets in the colex ordering of N ( r ) . Thefollowing result in [17] states that the value of l ( C r , m ) can be easily figured out when m is in a certain range. Qingsong Tang et al.
Lemma 2.1 (See [17], Lemma 2.4 )
For any integers m , t , and r satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) , we have l ( C r , m ) = l ([ t − ] ( r ) ) . Note that Conjectures 2.1 and 2.2 refine the following open conjecture of Frankl and F¨uredi.
Conjecture 2.3 (See [11], Conjecture 4.1) The r -graph with m edges formed by taking the first m sets in the colexordering of N ( r ) has the largest Lagrangian of all r -graphs with m edges. In particular, the r -graph with (cid:0) tr (cid:1) edgesand the largest Lagrangian is [ t ] ( r ) .Note that the upper bound (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) in Conjecture 2.1 is the best possible. For example, if m = (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) + l ( C r , m ) > l ([ t − ] ( r ) ) . To see this, take x : = ( x , . . ., x t ) ∈ S , where x = x = · · · = x t − = t − and x t − = x t = ( t − ) , then l ( C r , m ) ≥ l ( C r , m , x ) > l ([ t − ] ( r ) ) . In [17], Talbot proved the following.
Theorem 2.3 (See [17], Theorem 2.1)
Let m and t be integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − ( t − ) . Then l ( G ) ≤ l ([ t − ] ( ) ) . Theorem 2.4 (See [17], Theorem 3.1)
For any r ≥ there exists constants g r and k ( r ) such that if m satisfies (cid:18) t − r (cid:19) ≤ m ≤ (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − g r ( t − ) r − with t ≥ k ( r ) and G is an r-graph on t vertices with m edges, then l ( G ) ≤ l ([ t − ] ( r ) ) . Note that, Theorems 2.3 and 2.4 in this paper are equivalent to Theorems 2.1 and 3.1 in [18] after shifting t to t − Theorem 2.5 (See [19], Theorem 1.10) (a) Let m and t be positive integers satisfying (cid:18) t − r (cid:19) ≤ m ≤ (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − ( r − − )( (cid:18) t − r − (cid:19) − ) . Let G be an r-graph on t vertices with m edges and contain a clique of order t − . Then l ( G ) = l ([ t − ] ( r ) ) .(b) Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − ( t − ) . Let G be a -graph with medges and without containing a clique of order t − . Then l ( G ) < l ([ t − ] ( ) ) . In this paper, we provide upper bounds on the Lagrangian of a 3-graph, a 4-graph, and an r-graph, respectively,when the hypergraph contains dense subgraphs and the number of edges of the hypergraph is in a certain range.These results support Conjectures 2.1, 2.2 and extend Theorem 2.3.We will impose one additional condition on any optimal weighting x : = ( x , x , . . ., x n ) for an r -graph G : |{ i : x i > }| is minimal , i . e . if y is a feasible weighting for G satisfying |{ i : y i > }| < |{ i : x i > }| , then l ( G , y ) < l ( G ) . (1)When the theory of Lagrange multipliers is applied to find the optimum of l ( G ) , subject to (cid:229) ni = x i =
1, notethat l ( E i , x ) corresponds to the partial derivative of l ( G , x ) with respect to x i . The following lemma gives somenecessary conditions of an optimal weighting of l ( G ) . Lemma 2.2 (See [12], Theorem 2.1)
Let G : = ( V , E ) be an r-graph on the vertex set [ n ] and x : = ( x , x , . . ., x n ) be an optimal feasible weighting for G with k ( ≤ n) non-zero weights x , x , . . ., x k satisfying condition (1). Thenfor every { i , j } ∈ [ k ] ( ) , (a) l ( E i , x ) = l ( E j , x ) = r l ( G ) , (b) there is an edge in E containing both i and j. n Lagrangians of Hypergraphs Containing Dense Subgraphs 5 The following definition is also needed.
Definition 2.2 An r -graph G : = ( V , E ) on the vertex set [ n ] is left-compressed if j j · · · j r ∈ E implies i i · · · i r ∈ E provided i p ≤ j p for every p , ≤ p ≤ r . Equivalently, an r -graph G : = ( V , E ) is left-compressed iff E j \ i = /0 for any 1 ≤ i < j ≤ n . Remark 2.2 (a) In Lemma 2.2, part(a) implies that x j l ( E i j , x ) + l ( E i \ j , x ) = x i l ( E i j , x ) + l ( E j \ i , x ) . In particular, if G is left-compressed, then ( x i − x j ) l ( E i j , x ) = l ( E i \ j , x ) for any i , j satisfying 1 ≤ i < j ≤ k since E j \ i = /0.(b) If G is left-compressed, then for any i , j satisfying 1 ≤ i < j ≤ k , x i − x j = l ( E i \ j , x ) l ( E i j , x ) (2)holds. If G is left-compressed and E i \ j = /0 for i , j satisfying 1 ≤ i < j ≤ k , then x i = x j .(c) By (2), if G is left-compressed, then an optimal feasible weighting x : = ( x , x , . . ., x n ) for G must satisfy x ≥ x ≥ . . . ≥ x n ≥ . (3)In the proofs of our results, we need to consider various left-compressed 3-graphs on vertex set [t], which canbe obtained from a Hessian diagram as follows.A triple i i i is called a descendant of a triple j j j iff i s ≤ j s for each 1 ≤ s ≤
3, and i + i + i < j + j + j . In this case, the triple j j j is called an ancestor of i i i . The triple i i i is called a direct descendant of j j j if i i i is a descendant of j j j and j + j + j = i + i + i +
1. We say that j j j has lower hierarchy than i i i if j j j is an ancestor of i i i . This is a partial order on the set of all triples. Fig.1is a Hessian diagram on all triples on vertex set [ t ] . In this diagram, i i i and j j j are connected by an edge if i i i is a direct descendant of j j j . Remark 2.3
A 3-graph G is left-compressed iff all descendants of an edge of G are edges of G . Equivalently, if atriple is not an edge of G , then none of its ancestors will be an edge of G . -graphs Containing Subgraph K ( ) − t − Let K ( ) − t − denote the hypergraph obtained by K ( ) t − with one edge removed, where K ( ) t − stands for a complete3-graph with t − l − ( m , t − ) : = max { l ( G ) : G is a 3-graph with m edges and G containing K ( ) − t − but not containing K ( ) t − } . We now prove Theorem 3.1. Theorem 3.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Let G be a -graph with medges containing K ( ) − t − but not containing K ( ) t − . Then l ( G ) < l ([ t − ] ( ) ) for t ≥ . In the proof of Theorem 3.1, we need several lemmas.
Qingsong Tang et al. ... ... ... ... ...
Fig. 1: Hessian Diagram on [ t ] ( ) Lemma 3.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Then there exists a left-compressed -graph G with m edges containing [ t − ] ( ) \{ ( t − )( t − )( t − ) } but not containing [ t − ] ( ) such that l ( G ) = l ( ) − ( m , t − ) and there exists an optimal weighting x : = ( x , x , . . ., x n ) of G satisfying x i ≥ x j wheni < j. The proof of Lemma 3.1 is similar to the proof of Lemma 3.1 in [20]. However Lemma 3.1 in [20] cannot beused directly here. For completeness, we give the proof in Sect. 6.
Lemma 3.2 (See [20], Proposition 3.7 )
Let G be a -graph on t vertices with at most (cid:0) t − (cid:1) + (cid:0) t − (cid:1) edges. If Gdoes not contain K ( ) t − , then l ( G ) < l ([ t − ] ( ) ) for ≤ t ≤ . Lemma 3.3
Let G be a left-compressed 3-graph containing [ t − ] ( ) \{ ( t − )( t − )( t − ) } but not containing [ t − ] ( ) with m edges such that l ( G ) = l − ( m , t − ) . Let x : = ( x , x , . . ., x n ) be an optimal weighting of G and k bethe number of positive weights in x , then l ( G ) < l ([ t − ]) ( ) or | [ k − ] ( ) \ E | ≤ k − . The proof of Lemma 3.3 is similar to Lemma 3.2 in [20]. However Lemma 3.2 in [20] cannot be used directlyhere. For completeness, we give the details of the proof in Sect. 6.
Proof of Theorem 3.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Let G : = ( V , E ) be a3-graph with m edges containing K ( ) − t − but not containing K ( ) t − such that l ( G ) = l − ( m , t − ) . Let x : = ( x , x , . . ., x n ) be an optimal weighting of G and k be the number of non-zero weights in x . By Lemma 3.1,we can assume that G is left-compressed and contains [ t − ] ( ) \{ ( t − )( t − )( t − ) } but not contain [ t − ] ( ) and x ≥ x ≥ . . . ≥ x k > x k + = . . . = x n =
0. Since x has only k positive weights, we can assume that G is on [ k ] .Now we proceed to show that l ( G ) < l ([ t − ] ( ) ) . By Lemma 3.2, Theorem 3.1 holds when t ≤
12. Next weassume t ≥
13. If l ( G ) ≥ l ([ t − ] ( ) ) , then k ≥ t . Otherwise k ≤ t −
1, since G does not contain [ t − ] ( ) , then l ( G ) < l ([ t − ] ( ) ) .Since G is left-compressed and 1 ( k − ) k ∈ E , then | [ k − ] ( ) ∩ E k | ≥
1. If k ≥ t +
1, then applying Lemma3.3, we have m = | E | = | E ∩ [ k − ] ( ) | + | [ k − ] ( ) ∩ E k | + | E ( k − ) k | n Lagrangians of Hypergraphs Containing Dense Subgraphs 7 ≥ (cid:18) t (cid:19) − ( t − ) + ≥ (cid:18) t − (cid:19) + (cid:18) t − (cid:19) + , (4)which contradicts to the assumption that m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Recall that k ≥ t , so we have k = t . Since l − ( m , t − ) does not decrease as m increases, it is sufficient to show the case that m = (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Let G ′ : = G S { ( t − )( t − )( t − ) }\{ ( t − ) t } . If we can prove that l ( G , x ) < l ( G ′ , x ) , then since G ′ contains [ t − ] ( ) and G ′ has (cid:0) t − (cid:1) + (cid:0) t − (cid:1) edges, we have l ( G ′ , x ) ≤ l ( G ′ ) = l ([ t − ] ( ) ) . Consequently, l ( G ) < l ([ t − ] ) . Now we show that l ( G , x ) < l ( G ′ , x ) . Note that l ( G ′ , x ) − l ( G , x ) = x t − x t − x t − − x x t − x t . (5)By Remark 2.2(b), we have x = x t − + l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) , (6)and x t − = x t + l ( E ( t − ) \ t , x ) l ( E ( t − ) t , x ) . (7)Combining equations (5), (6) and (7), we get l ( G ′ , x ) − l ( G , x ) = x t − ( x t + l ( E ( t − ) \ t , x ) l ( E ( t − ) t , x ) ) x t − − ( x t − + l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) ) x t − x t = l ( E ( t − ) \ t , x ) l ( E ( t − ) t , x ) x t − x t − − l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) x t − x t . (8)By Remark 2.2(b) x = x t − + l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) ≤ x t − + x t − x t − + ( x + · · · + x t − ) x t x + · · · + x t − + x t < x t − + x t − + x t . (9)Hence l ( E ( t − ) , x ) − l ( E ( t − ) t , x ) ≥ x t − + x t − + x t − x > l ( E ( t − ) , x ) > l ( E ( t − ) t , x ) . Clearly x t − > x t since ( t − )( t − ) ∈ E ( t − ) \ t . Therefore to show that l ( G , x ) < l ( G ′ , x ) , it is sufficient to show that l ( E ( t − ) \ t , x ) ≥ l ( E \ ( t − ) , x ) . (10)If ( t − )( t − ) t ∈ E , then all triples in [ t ] ( ) \ { ( t − )( t − )( t − ) , i jt , where t − ≤ i < j ≤ t − } areedges in G since G is left-compressed. If E = [ t ] ( ) \ { ( t − )( t − )( t − ) , i jt , where t − ≤ i < j ≤ t − } , then m > (cid:0) t (cid:1) − ≥ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) (recall that t ≥ E = [ t ] ( ) \ { ( t − )( t − )( t − ) , i jt , where t − ≤ i < j ≤ t − } or ( t − )( t − ) t / ∈ E .If E = [ t ] ( ) \ { ( t − )( t − )( t − ) , i jt , where t − ≤ i < j ≤ t − } , then l ( E ( t − ) \ t , x ) = x t − x t − + x t − x t − + x t − x t − + x t − x t − + x t − x t − , and l ( E \ ( t − ) , x ) = x t − x t − + x t − x t + x t − x t + x t − x t + x t − x t . Qingsong Tang et al.
Clearly (10) holds in this case.If ( t − )( t − ) t / ∈ E , then l ( E ( t − ) \ t , x ) ≥ x t − l ( E ( t − )( t − ) ∩ E c ( t − ) t , x ) + x t − x t − + x t − x t − + x t − x t − = x t − l ( E c ( t − ) t , x ) + x t − x t − + x t − x t − + x t − x t − − x t − x t − − x t − x t − ≥ x t − l ( E c ( t − ) t , x ) + x t − x t − + x t − x t − − x t − x t − = x t − ( l ( E c ( t − ) t , x ) − x t − ) + x t − x t − + x t − x t − , and l ( E \ ( t − ) , x ) = x t l ( E c ( t − ) t , x ) + x t − x t − = x t ( l ( E c ( t − ) t , x ) − x t − ) + x t − x t − + x t − x t . Clearly (10) holds in this case.This completes the proof of Theorem 3.1. ⊓⊔ Remark 3.1
Note that for t ≤
5, the left-compressed 3-graph with (cid:0) t − (cid:1) + (cid:0) t − (cid:1) edges always contains K ( ) t − .Combining Theorems 2.5 and 3.1, we have that, if G is a 3-graph containing K ( ) − t − with at most (cid:0) t − (cid:1) + (cid:0) t − (cid:1) edges, then l ( G ) ≤ l ([ t − ] ( ) ) .Also, applying Theorem 3.1, we derive two easy corollaries that support Conjecture 2.2. Corollary 3.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Let G : = ( V , E ) be a left-compressed 3-graph on the vertex set [t] with m edges and not containing a clique of order t − . If | E ( t − ) t | ≤ ,then l ( G ) < l ([ t − ] ( ) ) .Proof Because l − ( m , t − ) doesn’t decrease as m increases, we can assume that m = (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Since G : = ( V , E ) does not contain [ t − ] ( ) and G is left-compressed, then ( t − )( t − )( t − ) / ∈ E . If | E ( t − ) t | = G must contain [ t − ] ( ) . Therefore, | E ( t − ) t | = t ≤
5, Theorem 3.1 clearly holds. Next, we assume t ≥ | E ( t − ) t | =
2. Note that G is left-compressed, in view of Fig.1, E = [ t ] ( ) \ { ( t − ) t , ( t − ) t , · · · ( t − )( t − ) t , ( t − )( t − )( t − ) , ( t − )( t − ) t } . Case 2. | E ( t − ) t | =
3. In this case, since G is left-compressed, in view of Fig.1, we only need to consider E =[ t ] ( ) \ { ( t − ) t , · · · ( t − )( t − ) t , ( t − )( t − )( t − ) , ( t − )( t − ) t , ( t − )( t − ) t } .In both cases, left-compressed 3-graph G does not contain the edge (t-3)(t-2)(t-1). Thus, the conditions inTheorem 3.1 are satisfied. Therefore, we are done. ⊓⊔ The next corollary states that if 3-graph G contains a dense subgraph close to the structure in C , m , then wehave l ( G ) < l ([ t − ] ( ) ) . Corollary 3.2
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) . Let G : = ( V , E ) be a left-compressed 3-graph on the vertex set [t] with m edges and not containing a clique of size t − , and | E ( G ) D E ( C , m ) | ≤ . Then, l ( G ) < l ([ t − ] ( ) ) .Proof If m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) , then | E ( t − ) t | ≤
3, since otherwise | E ( G ) D E ( C , m ) | >
6. Applying Corollary 3.2, wehave l ( G ) < l ([ t − ] ( ) ) . ⊓⊔ n Lagrangians of Hypergraphs Containing Dense Subgraphs 9 t − or t − Theorem 4.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − t − . Let G be a -graph withm edges and G contain the maximum clique of order t − . Then l ( G ) < l ([ t − ] ( ) ) . Theorem 4.2
Let m and t be positive integers satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) − r − ( (cid:0) t − r − (cid:1) − ) . Let G bean r-graph on t vertices with m edges and with a clique of order t − . Then l ( G ) ≤ l ([ t − ] ( r ) ) . Theorem 4.3
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) ⌊ t − ⌋ (cid:1) . Let G be a -graph with medges and a clique of order t − . Then l ( G ) = l ([ t − ] ( ) ) . Here (cid:0) t − (cid:1) + (cid:0) ⌊ t − ⌋ (cid:1) is not the best upper bound that we can obtain. This bound is for simplicity of the proof.Denote l r ( m , p ) : = max { l ( G ) : G is an r − graph with m edges and G contains a maximum clique of order p } . Similar to the proof of Lemma 3.1 in [20], we can prove the following lemma. We will give the proof in Sect.6.
Lemma 4.1
Let m and t be positive integers satisfying (cid:18) t − (cid:19) ≤ m ≤ (cid:18) t − (cid:19) + (cid:18) t − (cid:19) − t − . Then there exists a left-compressed -graph G with m edges containing the maximum clique [ t − ] ( ) such that l ( G ) = l ( m , t − ) . Similar to the proof of Lemma 3.2 in [20], we have the following lemma. For completeness, we will give theproof in Sect. 6.
Lemma 4.2
Let G be a left-compressed 3-graph containing the maximum clique [ t − ] ( ) with m edges such that l ( G ) = l ( m , t − ) . Let x : = ( x , x , . . ., x n ) be an optimal weighting of G and k be the number of positive weights in x , then l ( G ) < l ([ t − ]) ( ) or | [ k − ] ( ) \ E | ≤ k − . We also need the following lemma whose proof is similar to Lemma 2.7 in [17] and Lemma 3.3 in [16]. Wewill give it in Sect. 6.
Lemma 4.3
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − t − . Let G be a left-compressed3-graph on the vertex set [ t ] and contain the maximum clique [ t − ] ( ) with m edges such that l ( G ) = l ( m , t − ) .Assume b : = | E ( t − ) t | , then l ( G ) < l ([ t − ] ( ) ) or | [ t − ] ( ) \ E t | ≤ b . Proof of Theorem 4.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − t − . Clearly wecan assume that t ≥
5. Let G : = ( V , E ) be a 3-graph with m edges containing a maximum clique of order t − l ( G ) = l ( m , t − ) . Let x : = ( x , x , . . ., x n ) be an optimal weighting of G and k be the number of non-zeroweights in x . By Lemma 4.1, we can assume that G is left-compressed with the maximum clique [ t − ] ( ) and x ≥ x ≥ . . . ≥ x k > x k + = . . . = x n =
0. Since x has only k positive weights, we can assume that G is on [ k ] .Now we proceed to show that l ( G ) < l ([ t − ] ( ) ) . If l ( G ) ≥ l ([ t − ] ( ) ) , then k ≥ t . Otherwise k ≤ t − G does not contain [ t − ] ( ) , then l ( G ) < l ([ t − ] ( ) ) . By Lemma 2.2(a), k − k appear in somecommon edge e ∈ E . Recall that E is left-compressed, so 1 ( k − ) k ∈ E . Define b : = max { i : i ( k − ) k ∈ E } . Because E is left-compressed, E i \ j = /0 for 1 ≤ i < j ≤ b . Hence, by Remark 2.2(a), we have x = x = · · · = x b .Clearly, b ≤ k − G is left-compressed and 1 ( k − ) k ∈ E , then | [ k − ] ( ) ∩ E k | ≥
1. So applying Lemma 4.2, similar to(4), we have k = t .Since k = t , we can assume that G is on [ t ] . By Remark 2.2(b), we have x = x t − + l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) . Recall that G contains a clique order of t −
2, we have l ( E \ ( t − ) , x ) = x t − l ( E c ( t − )( t − ) , x ) + x t l ( E c ( t − ) t , x ) − x t − x t . Hence x < x t − + x t − l ( E c ( t − )( t − ) , x ) + x t l ( E c ( t − ) t , x ) l ( E ( t − ) , x ) . Since for i = t − , t − , t − i ∈ E c ( t − ) t implies that i ( t − ) ∈ [ t − ] ( ) \ E t and i ∈ E c ( t − ) t implies that i ( t − ) ∈ [ t − ] ( ) \ E t , t − ∈ E c ( t − ) t , t − ∈ E c ( t − ) t , and t − ∈ E c ( t − ) t , t − ∈ E c ( t − ) t and ( t − )( t − ) ∈ [ t − ] ( ) \ E t , applying Lemma 4.3, then | E c ( t − ) t | + | E c ( t − ) t | ≤ | [ t − ] ( ) \ E t | + ≤ b + . Note that b ≤ t − | E c ( t − ) t | ≤ | E c ( t − ) t | , So | E c ( t − ) t | ≤ b + ≤ t − . Since G is left-compressed, then | E c ( t − )( t − ) | ≤ | E c ( t − ) t | ≤ t − . So x < x t − + x t − l ( E c ( t − )( t − ) , x ) + x t l ( E c ( t − ) t , x ) l ( E ( t − ) , x ) ≤ x t − + x t − t − l ( E ( t − ) , x ) t − + x t t − l ( E ( t − ) , x ) t − l ( E ( t − ) , x ) ≤ x t − . This implies 2 x t − x t − x t − − x x t − x t > . Let C : = [ t − ] ( ) \ E be all triples containing t − E , E ′ : = E S C \{ ( b − ⌊ | C | ⌋ + )( t − ) t , ( b − ⌊ | C | ⌋ + )( t − ) t , . . ., b ( t − ) t } and G ′ : = ([ t ] ( ) , E ′ ) . Then l ( G ′ , x ) − l ( G , x ) = l ( C , x ) − ⌊ | C | ⌋ x x t − x t ≥ | C | x t − x t − x t − − ⌊ | C | ⌋ x x t − x t ≥ | C | ( x t − x t − x t − − x x t − x t ) > . n Lagrangians of Hypergraphs Containing Dense Subgraphs 11 So l ( G , x ) < l ( G ′ , x ) . Because | E ′ | = | E | + | C | − ⌊ | C | ⌋ ≤ | E | + | C | + ≤ (cid:18) t − (cid:19) + (cid:18) t − (cid:19) − t − + t − + = (cid:18) t − (cid:19) + (cid:18) t − (cid:19) . and G ′ contains a clique of order t −
1, we have l ( G ′ , x ) ≤ l ( G ′ ) = l ([ t − ] ( ) ) by Theorem 2.2. Hence l ( G , x ) < l ( G ′ , x ) ≤ l ([ t − ] ( ) ) . This proves Theorem 4.1. ⊓⊔ The following lemma implies that we only need to consider left-compressed r -graphs when Theorem 4.2 isproved. The proof is given in Sect. 6. Lemma 4.4
Let m and t be positive integers satisfying (cid:18) t − r (cid:19) ≤ m ≤ (cid:18) tr (cid:19) − . Then there exists a left-compressed G with m edges containing the clique [ t − ] ( r ) such that l ( G ) = l r ( m , t − ) andthere exists an optimal weighting x : = ( x , x , . . ., x n ) of G satisfying x i ≥ x j when i < j. We also need the following in the proof of Theorem 4.2 and Theorem 4.3
Lemma 4.5 (See [19], Theorem 3.4)
Let r ≥ and t ≥ r + be positive integers. Let G be a left-compressedr-graph on t vertices satisfying | [ t − ] ( r − ) \ E t | ≥ r − | E ( t − ) t | . Then(a) If G contains [ t − ] ( r ) , then l ( G ) = l ([ t − ] ( r ) ) ,(b) If G does not contain [ t − ] ( r ) , then l ( G ) < l ([ t − ] ( r ) ) .Proof of Theorem 4.2 Let m and t be positive integers satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) t − r (cid:1) + (cid:0) t − r − (cid:1) − r − ( (cid:0) t − r − (cid:1) − ) .Let G be an r -graph with m edges and t vertices with a clique order of t −
2. By Lemma 4.4 we can assume G is left-compressed. By Lemma 4.5, it is sufficient to show that | [ t − ] ( r − ) \ E t | ≥ r − | E ( t − ) t | . If not, then | [ t − ] ( r − ) \ E t | < r − | E ( t − ) t | and | [ t − ] ( r − ) \ E t − | ≤ | [ t − ] ( r − ) \ E t | < r − | E ( t − ) t | . Since G contains theclique [ t − ] ( r ) , then m = (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − | [ t − ] ( r − ) \ E t | − | [ t − ] ( r − ) \ E t − | + | E ( t − ) t | > (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − (cid:18) t − r − (cid:19) − ( r − − ) | E ( t − ) t | + ≥ (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − r − ( (cid:18) t − r − (cid:19) − ) . since | E ( t − ) t | ≤ (cid:0) t − r − (cid:1) −
1, this is a contradiction. Note that, if | E ( t − ) t | = (cid:0) t − r − (cid:1) , then E = [ t ] ( r ) since G is left-compressed and m = (cid:0) tr (cid:1) , which results in a contradiction too. This proves Theorem 4.2. ⊓⊔ Remark 4.1
Lemma 4.5(b) and Theorem 4.2 imply that if m and t are positive integers satisfying (cid:18) t − r (cid:19) ≤ m ≤ (cid:18) t − r (cid:19) + (cid:18) t − r − (cid:19) − r − ( (cid:18) t − r − (cid:19) − ) and G is a r -graph on t vertices with m edges and with a maximum clique of order t −
2. Then l ( G ) < l ([ t − ] ( r ) ) . Proof of Theorem 4.3
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) ⌊ t − ⌋ (cid:1) . Let G be a 4-graphwith m edges and a clique of order t −
1. Since it contains a clique of order t-1, without loss of generality, we mayassume that it contains [ t − ] ( ) . Since G contains [ t − ] ( ) , we have l ( G ) ≥ l ([ t − ] ( ) ) . Next we prove that l ( G ) ≤ l ([ t − ] ( ) ) . Let x : = ( x , x , . . ., x n ) be an optimal weighting of G and k be the number of non-zero weights in x . If k ≤ t − l ( G ) ≤ l ([ t − ] ( ) ) . Assume that k ≥ t . Recall that (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) ⌊ t − ⌋ (cid:1) and G contains [ t − ] ( ) ,hence | E k | ≤ (cid:0) ⌊ t − ⌋ (cid:1) . By Fact 2.1, Lemma 2.2(a) and Theorem 2.3, we have l ( G , x ) = l ( E k , x ) ≤ (cid:18) ⌊ t − ⌋ (cid:19) ( ⌊ t − ⌋ ) ≤ ( t − )( t − ) ( t − ) < ( t − )( t − )( t − ) ( t − ) = l ([ t − ] ( ) ) . ⊓⊔ Remark 4.2
Also, note that Theorem 3.1, Theorem 4.1, and Remark 4.1 provide further evidence for Conjecture2.2. Theorem 4.3 provide further evidence for Conjecture 2.1.
Frankl and F¨uredi [11] asked the following question: Given r ≥ m ∈ N how large can the Lagrangian of an r -graph with m edges be? Conjecture 2.3 proposes a solution to the question mentioned above.Denote l rm : = max { l ( G ) : G is an r − graph with m edges } . The following lemma implies that we only need to consider left-compressed r -graphs when Conjecture 2.3 isexplored. Lemma 5.1 (See [17], Lemma 2.3)
There exists a left-compressed r-graph G with m edges such that l ( G ) = l rm . We extend Theorem 2.3 in Theorem 5.1 which is a corollary of Theorem 3.1.
Theorem 5.1
Let m and t be positive integers satisfying (cid:0) t − (cid:1) ≤ m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) − ( t − ) . Then Conjecture 2.3is true for r = and this value of m.Proof Let x : = ( x , x , . . ., x n ) be an optimal weighting for G and k be the number of positive weights in x . We canassume that G is left-compressed by Lemma 5.1. So x ≥ x ≥ . . . ≥ x k > x k + = . . . = x n = x has only k positive weights, we can assume that G is on vertex set [ k ] .Now we proceed to show that l ( G ) ≤ l ([ t − ] ( ) ) . If l ( G ) > l ([ t − ] ( ) ) , then k ≥ t since otherwise k ≤ t − l ( G ) ≤ l ([ t − ] ( ) ) . Next we apply the following lemma. Lemma 5.2 (See [17], Lemma 2.5) . Let m be a positive integer. Let G be a left-compressed 3-graph with m edgessuch that l ( G ) = l m . Let x : = ( x , x , . . ., x n ) be an optimal weighting for G and k be the number of non-zeroweights in x , then | [ k − ] ( ) \ E | ≤ k − . So similar to (4), we have k = t . Next we need the following lemma whose proof follows the lines of Lemma 2.5in [17]. For completeness, we give the proof in Sect. 6. n Lagrangians of Hypergraphs Containing Dense Subgraphs 13 Lemma 5.3
Let G be a left-compressed 3-graph on the vertex set [ t ] with m edges where (cid:18) t − (cid:19) ≤ m ≤ (cid:18) t − (cid:19) + (cid:18) t − (cid:19) , and l ( G ) = l m . Let x : = ( x , x , . . ., x t ) be an optimal weighting for G. Then | [ t − ] ( ) \ E | ≤ t − , or l ( G ) ≤ l ([ t − ] ( ) ) . Assume Lemma 5.3 holds, we continue the proof of Theorem 5.1. If l ( G ) > l ([ t − ] ( ) ) , then | [ t − ] ( ) \ E | ≤ t − | [ t − ] ( ) \ E | − [ t − ] ( ) \ E to E and let the new3-graph be G ′ . Then G ′ contains K ( ) − t − , the number of edges in G ′ is at most (cid:0) t − (cid:1) + (cid:0) t − (cid:1) and l ( G ′ ) ≥ l ( G ) .Applying Theorem 2.5 and Theorem 3.1, l ( G ′ ) ≤ l ([ t − ] ( ) ) . Therefore l ( G ) ≤ l ([ t − ] ( ) ) = l ( C , m ) byLemma 2.1. This completes the proof of Theorem 5.1. ⊓⊔ Proof techniques of lemmas in this section follow from proof techniques of some lemmas in [17, 19, 20]. Asmentioned earlier, lemmas in those papers cannot be applied directly to situations in this paper. For completeness,we give the proof of these lemmas in this section.
Proof of Lemma 3.1
Let G be a 3-graph on the vertex set [ n ] with m edges containing K ( ) − t − but not containing K ( ) t − such that l ( G ) = l − ( m , t − ) . We call such a 3-graph G an extremal 3-graph for m and t −
1. Let x : = ( x , x , . . ., x n ) be an optimal weighting of G . We can assume that x i ≥ x j when i < j since otherwise wecan just relabel the vertices of G and obtain another extremal 3-graph for m and t − x : = ( x , x , . . ., x n ) satisfying x i ≥ x j when i < j . Next we obtain a new 3-graph G ′ from G by performing thefollowing:1. If ( t − )( t − )( t − ) ∈ E ( G ) , then there is at least one triple in [ t − ] ( ) \ E ( G ) , we replace ( t − )( t − )( t − ) by this triple;2. If an edge in G has a descendant other than ( t − )( t − )( t − ) that is not in E ( G ) , then replace this edge bya descendant other than ( t − )( t − )( t − ) with the lowest hierarchy. Repeat this until there is no such anedge.Then G ′ satisfies the following properties:1. The number of edges in G ′ is the same as the number of edges in G .2. l ( G ) = l ( G , x ) ≤ l ( G ′ , x ) ≤ l ( G ′ ) . ( t − )( t − )( t − ) / ∈ E ( G ′ ) .4. [ t − ] ( ) \{ ( t − )( t − )( t − ) } ∈ E ( G ′ ) .5. For any edge in E ( G ′ ) , all its descendants other than ( t − )( t − )( t − ) will be in E ( G ′ ) .If G ′ is not left-compressed, then there is an ancestor uvw of ( t − )( t − )( t − ) such that uvw ∈ E ( G ′ ) .We claim that uvw must be ( t − )( t − ) t . If uvw is not ( t − )( t − ) t , then since all descendants other than ( t − )( t − )( t − ) of uvw will be in E ( G ′ ) , then all descendants of ( t − )( t − ) t (other than ( t − )( t − )( t − ) ) or all descendants of ( t − )( t − )( t + ) (other than ( t − )( t − )( t − ) ) will be in E ( G ′ ) .So all triples in [ t − ] ( ) \ { ( t − )( t − )( t − ) } , all triples in the form of i jt (where i j ∈ [ t − ] ( ) ), and all triplesin the form of i j ( t + ) (where i j ∈ [ t − ] ( ) ) or all triples in the form of i ( t − ) l , 1 ≤ i ≤ t − E ( G ′ ) ,then m ≥ (cid:18) t − (cid:19) − + (cid:18) t − (cid:19) + ( t − ) > (cid:18) t − (cid:19) + (cid:18) t − (cid:19) , which is a contradiction. So uvw must be ( t − )( t − ) t . Since m ≤ (cid:0) t − (cid:1) + (cid:0) t − (cid:1) and all the descendants otherthan ( t − )( t − )( t − ) of an edge in G ′ will be an edge in G ′ , then there are two possibilities.Case 1. E ( G ′ ) = ([ t − ] ( ) \ { ( t − )( t − )( t − ) } ) ∪ { i jt , i j ∈ [ t − ] ( ) } ∪ { ( t + ) } . Case 2. E ( G ′ ) = ([ t − ] ( ) \ { ( t − )( t − )( t − ) } ) ∪ { i jt , i j ∈ [ t − ] ( ) } . Let y : = ( y , y , . . ., y n ) be an optimal weighting of G ′ , where n = t + n = t . We claim that if Case 1happens, then y t = y t + =
0, since E ( t − ) t = E t ( t + ) = /0(by Lemma 2.2). If Case 2 happens, then y t = E ( t − ) t = f (by Lemma 2.2). Hence we can assume that G is left-compressed. ⊓⊔ Proof of Lemma 3.3
Since G contains the clique of [ t − ] ( ) \{ ( t − )( t − )( t − ) } , it is true for k ≤ t . Next weassume that k ≥ t + G is left-compressed, 1 ( k − ) k ∈ E . Let b : = max { i : i ( k − ) k ∈ E } . Since E is left-compressed, then E i : = { , . . ., i − , i + , . . ., k } ( ) , for 1 ≤ i ≤ b , and E i \ j = /0 for 1 ≤ i < j ≤ b . Hence, by Remark 2.2(a), we have x = x = · · · = x b .We define a new feasible weighting y for G as follows. Let y i = x i for i = k − , k , y k − = x k − + x k and y k = l ( E k − , x ) = l ( E k , x ) , so l ( G , y ) − l ( G , x ) = x k ( l ( E k − , x ) − x k l ( E k ( k − ) , x )) − x k ( l ( E k , x ) − x k − l ( E k ( k − ) , x )) − x k − x k l ( E k ( k − ) , x )= x k ( l ( E k − , x ) − l ( E k , x )) − x k b (cid:229) i = x i = − bx x k . (11)Since y k = k from E to form a new 3-graph G : = ([ k ] , E ) with | E | : = | E | − | E k | and l ( G , y ) = l ( G , y ) . We will show that if Lemma 3.3 fails to hold then there exists a set ofedges F ⊂ [ k − ] ( ) \ E satisfying l ( F , y ) > bx x k (12)and | F | ≤ | E k | . (13)Then, using (11), (12), and (13), the 3-graph G ′ : = ([ k ] , E ′ ) , where E ′ : = E ∪ F , satisfies | E ′ | ≤ | E | and l ( G ′ , y ) = l ( G , y ) + l ( F , y ) > l ( G , y ) + bx x k = l ( G , x ) . Hence l ( G ′ ) > l ( G ) . Note that G ′ still contains [ t − ] ( ) \{ ( t − )( t − )( t − ) } since G ′ contains all edges in E ∩ [ k − ] ( ) ⊇ E ∩ [ t − ] ( ) . If G ′ does not contains a clique of size t −
1, note that G ′ still contain [ t − ] ( ) \{ ( t − )( t − )( t − ) } , it contradicts to l ( G ) = l − ( m , t − ) . If G ′ contains a clique of size t −
1, then byTheorem 2.5 l ( G ′ ) = l ([ t − ] ( ) ) and consequently l ( G ′ ) < l ([ t − ] ( ) ) .We must now construct the set of edges F satisfying (12) and (13). Applying Remark 2.2(a) by taking i = , j = k −
1, we have x = x k − + l ( E \ ( k − ) , x ) l ( E ( k − ) , x ) . Let C : = [ k − ] ( ) \ E k − . Then l ( E \ ( k − ) , x ) = x k (cid:229) k − i = b + x i + l ( C , x ) . Applying this and multiplying bx k to theabove equation (note that l ( E ( k − ) , x ) = (cid:229) ki = , i = k − x i ), we have bx x k = bx k − x k + bx k (cid:229) k − i = b + x i (cid:229) ki = , i = k − x i + bx k l ( C , x ) (cid:229) ki = , i = k − x i . n Lagrangians of Hypergraphs Containing Dense Subgraphs 15 Since x ≥ x ≥ · · · ≥ x k , then bx x k ≤ bx k − x k ( + k − ( b + ) k − ) + bx k l ( C , x ) k − . (14)Define a : = ⌈ b | C | k − ⌉ and b : = ⌈ b ( + k − ( b + ) k − ) ⌉ . Note that ⌈ b ( + k − ( b + ) k − ) ⌉ ≤ k − b ≤ k −
2. So b ≤ k − F ⊂ [ k − ] ( ) \ E consist of the a heaviest edges in [ k − ] ( ) \ E containing the vertex k − | [ k − ] ( ) \ E k − | = | C | ≥ a ). Recalling that y k − = x k − + x k we have l ( F , y ) ≥ bx k l ( C , x ) k − + a x k − x k . So using (14) l ( F , y ) − bx x k ≥ x k − x k ( a − b ) . (15)We now distinguish two cases.Case 1. a > b .In this case l ( F , y ) − bx k − x k > F : = F satisfies (12). We need to check that | F | ≤ | E k | . Since E is left-compressed, then [ b ] ( ) ∪ { , . . ., b } × { b + , . . . , k − } ⊂ E k . Hence | E k | ≥ b [ b − + ( k − − b )] ≥ b ( k − ) b ≤ k −
2. Recall that | F | = a = ⌈ b | C | k − ⌉ . Since C ⊂ [ k − ] ( ) , we have | C | ≤ (cid:0) k − (cid:1) . So using (16) we obtain | F | ≤ ⌈ b ( k − ) ⌉ ≤ b ( k − ) ≤ | E k | . So both (12) and (13) are satisfied.Case 2. a ≤ b .Suppose that Lemma 3.3 fails to hold. So | [ k − ] ( ) \ E | ≥ k − ≥ b + b ≤ k − F consistof any b + − a edges in [ k − ] ( ) \ ( E ∪ F ) and define F : = F ∪ F . Then since l ( F , y ) ≥ ( b + − a ) x k − andusing (15), l ( F , y ) − bx k − x k = l ( F , y ) − bx k − x k + l ( F , y ) ≥ ( b + − a ) x k − − x k − x k ( b − a ) > . So (12) is satisfied. What remains is to check that | F | ≤ | E k | . In fact, | F | = b + ≤ k − ≤ b ( k − ) ≤ | E k | when b ≥
2. If b =
1, then, | F | = b + = ≤ k − = b [ b − + ( k − − b )] ≤ | E k | since k ≥ t ≥ ⊓⊔ Proof of Lemma 4.1
Let G be a 3-graph on the vertex set [ n ] with m edges containing a maximal clique of order t − l ( G ) = l ( m , t − ) . We call such a G an extremal 3-graph for m and t −
2. Let x : = ( x , x , · · · , x n ) bean optimal weighting of G . We can assume that x i ≥ x j when i < j since otherwise we can just relabel the verticesof G and obtain another extremal 3-graph for m and t − x : = ( x , x , · · · , x n ) satisfying x i ≥ x j when i < j . Next we obtain a new 3-graph G ′ from G by performing the followings1. If ( t − )( t − )( t − ) ∈ E ( G ) , then there is at least one triple in [ t − ] ( ) \ E ( G ) , we replace ( t − )( t − )( t − ) by this triple;2. If an edge in G has a descendant other than ( t − )( t − )( t − ) that is not in E ( G ) , then replace this edgeby a descendant other than ( t − )( t − )( t − ) with the lowest hierarchy. Repeat this until there is no such anedge.Then G ′ satisfies the followings1. The number of edges in G ′ is the same as the number of edges in G ;2. G contains the clique [ t − ] ( ) ;3. l ( G ) = l ( G , x ) ≤ l ( G ′ , x ) ≤ l ( G ′ ) ;4. ( t − )( t − )( t − ) / ∈ E ( G ′ ) ;5. For any edge in E ( G ) , all its descendants other than ( t − )( t − )( t − ) will be in E ( G ′ ) .If G ′ is not left-compressed, then there is an ancestor uvw of ( t − )( t − )( t − ) such that uvw ∈ G ′ and allthe descendant of uvw other than uvw are in G ′ . Hence E ( G ′ ) ⊇ ([ t − ] ( ) \{ ( t − )( t − )( t − ) } ) ∪ { i jt , i j ∈ [ t − ] ( ) } .and m ≥ (cid:18) t − (cid:19) − + (cid:18) t − (cid:19) > (cid:18) t − (cid:19) + (cid:18) t − (cid:19) − t − . which is a contradiction. Hence G ′ is left-compressed. ⊓⊔ Proof of Lemma 4.2
Since G contains the clique of [ t − ] ( ) , it is true for k ≤ t −
1. Assume that k ≥ t .Since G is left-compressed, 1 ( k − ) k ∈ E . Let b : = max { i : i ( k − ) k ∈ E } . Since E is left-compressed, E i = { , . . ., i − , i + , . . . , k } ( ) , for 1 ≤ i ≤ b , and E i \ j = /0 for 1 ≤ i < j ≤ b . Hence, by Remark 2.2(a), we have x = x = · · · = x b .We define a new feasible weighting y for G as follows. Let y i : = x i for i = k − , k , y k − : = x k − + x k and y k : = l ( E k − , x ) = l ( E k , x ) , so l ( G , y ) − l ( G , x ) = x k ( l ( E k − , x ) − x k l ( E k ( k − ) , x )) − x k ( l ( E k , x ) − x k − l ( E k ( k − ) , x )) − x k − x k l ( E k ( k − ) , x )= x k ( l ( E k − , x ) − l ( E k , x )) − x k b (cid:229) i = x i = − bx x k . (17)Since y k = k from E to form a new 3-graph G : = ([ k ] , E ) with | E | : = | E | − | E k | and l ( G , y ) = l ( G , y ) . We will show that if Lemma 4.2 fails to hold then there exists a set ofedges F ⊂ [ k − ] ( ) \ E satisfying l ( F , y ) > bx x k , (18)and | F | ≤ | E k | . (19)Then, using (17), (18), and (19), the 3-graph G ′ : = ([ k ] , E ′ ) , where E ′ : = E ∪ F , satisfies | E ′ | ≤ | E | and l ( G ′ , y ) = l ( G , y ) + l ( F , y ) > l ( G , y ) + bx x k = l ( G , x ) . Hence l ( G ′ ) > l ( G ) . Note that G ′ still contains the clique [ t − ] ( ) since G ′ contains all edges in E ∩ [ k − ] ( ) ⊃ [ t − ] ( ) . If G ′ does not contains a clique of size t −
1, it contradicts to l ( G ) = l ( m , t − ) . If G ′ n Lagrangians of Hypergraphs Containing Dense Subgraphs 17 contains a clique of size t −
1, then by Theorem 2.2 l ( G ′ ) = l ([ t − ] ( ) ) and consequently l ( G ′ ) < l ([ t − ] ( ) ) .We must now construct the set of edges F satisfying (18) and (19). Applying Remark 2.2(a) by taking i = , j = k −
1, we have x = x k − + l ( E \ ( k − ) , x ) l ( E ( k − ) , x ) . Let C : = [ k − ] ( ) \ E k − . Then l ( E \ ( k − ) , x ) = x k (cid:229) k − i = b + x i + l ( C , x ) . Applying this and multiplying bx k to theabove equation (note that l ( E ( k − ) , x ) = (cid:229) ki = , i = k − x i ), we have bx x k = bx k − x k + bx k (cid:229) k − i = b + x i (cid:229) ki = , i = k − x i + bx k l ( C , x ) (cid:229) ki = , i = k − x i . Since x ≥ x ≥ · · · ≥ x k , then bx x k ≤ bx k − x k ( + k − ( b + ) k − ) + bx k l ( C , x ) k − . (20)Define a : = ⌈ b | C | k − ⌉ and b : = ⌈ b ( + k − ( b + ) k − ) ⌉ . Note that ⌈ b ( + k − ( b + ) k − ) ⌉ ≤ k − b ≤ k −
2. So b ≤ k − F ⊂ [ k − ] ( ) \ E consist of the a heaviest edges in [ k − ] ( ) \ E containing the vertex k − | [ k − ] ( ) \ E k − | = | C | ≥ a ). Recalling that y k − = x k − + x k we have l ( F , y ) ≥ bx k l ( C , x ) k − + a x k − x k . So using (20) l ( F , y ) − bx x k ≥ x k − x k ( a − b ) . (21)We now distinguish two cases.Case 1. a > b .In this case l ( F , y ) − bx k − x k > F : = F satisfies (18). We need to check that | F | ≤ | E k | . Since E is left-compressed, then [ b ] ( ) ∪ { , . . ., b } × { b + , . . . , k − } ⊂ E k . Hence | E k | ≥ b [ b − + ( k − − b )] ≥ b ( k − ) b ≤ k −
2. Recall that | F | = a = ⌈ b | C | k − ⌉ . Since C ⊂ [ k − ] ( ) , we have | C | ≤ (cid:0) k − (cid:1) . So using (20) we obtain | F | ≤ ⌈ b ( k − ) ⌉ ≤ b ( k − ) ≤ | E k | . So both (18) and (19) are satisfied.Case 2. a ≤ b .Suppose that Lemma 4.2 fails to hold. So | [ k − ] ( ) \ E | ≥ k − ≥ b + b ≤ k − F consistof any b + − a edges in [ k − ] ( ) \ ( E ∪ F ) and define F : = F ∪ F . Then since l ( F , y ) ≥ ( b + − a ) x k − andusing (21), l ( F , y ) − bx k − x k = l ( F , y ) − bx k − x k + l ( F , y ) ≥ ( b + − a ) x k − − x k − x k ( b − a ) > . So (18) is satisfied. What remains is to check that | F | ≤ | E k | . In fact, | F | = b + ≤ k − ≤ b ( k − ) ≤ | E k | when b ≥
2. If b =
1, then applying (21), | F | = b + = ≤ k − = b [ b − + ( k − − b )] ≤ | E k | since k ≥ t ≥ ⊓⊔ Proof of Lemma 4.3
Let b : = max { i : i ( t − ) t ∈ E } . Since E is left-compressed, then E i = { , . . . , i − , i + , . . ., t } ( ) , for 1 ≤ i ≤ b and E i \ j = /0 for 1 ≤ i < j ≤ b .Hence, by Remark 2.2(a), we have x = x = · · · = x b . Consider a new weighting for G , z : = ( z , z , . . ., z t ) given by z i : = x i for i = t − , t , z t − : = z t : = x t − + x t . By Lemma 2.2(a), l ( E t − , x ) = l ( E t , x ) , so l ( G , z ) − l ( G , x ) = x t − ( l ( E t , x ) − l ( E t − , x )) − x t − b (cid:229) i = x i = − bx x t − . (23)Since z t − = t − E to form a new 3-graph G : = ([ t ] , E ) with | E | : = | E | − | E t − | and l ( G , z ) = l ( G , z ) .If | [ t − ] ( ) \ E t | > b , we will show that there exists a set of edges F ⊂ { , ..., t − , t } ( ) \ E satisfying l ( F , z ) > bx x t − . (24)Then using (23) and (24), the 3-graph G ′ : = ([ t ] , E ′ ) , where E ′ : = E ∪ F , satisfies l ( G ′ , z )) > l ( G ) . Since z has only t − l ( G ′ , z )) ≤ l ([ t − ] ( ) ) , and consequently l ( G ) < l ([ t − ] ( ) ) . We must now construct the set of edges F . Since G is left-compressed, applying Remark 2.2(a) by taking i = j = t , we get x = x t + l ( E \ t , x ) l ( E t , x ) . Let D : = [ t − ] ( ) \ E t . Then l ( E \ t , x ) = x t − (cid:229) t − i = b + x i + l ( D , x ) . Applying this and multiplying bx t − to theabove equation (note that l ( E t , x ) = (cid:229) t − i = x i ), we have bx x t − = bx t x t − + bx t − (cid:229) t − i = b + x i (cid:229) t − i = x i + bx t − l ( D , x ) (cid:229) t − i = x i . Let c : = (cid:229) t − i = b + x i (cid:229) t − i = x i and d : = bx t − (cid:229) t − i = x i . Then bx x t − = bx t x t − + bcx t − + dx t − l ( D , x ) . (25)Let F consist of those edges in { , ..., t − , t } ( ) \ E containing the vertex t . Then l ( F , z ) = ( x t − + x l ) l ( D , x ) . (26)Since | [ t − ] ( ) \ E t | > b , then l ( D , x ) > bx t − . (27) n Lagrangians of Hypergraphs Containing Dense Subgraphs 19 Applying equations (25), (26), and (27), we get l ( F , z ) − bx x t − = ( x t − + x t ) l ( D , x ) − bx t x t − − bcx t − − dx t − l ( D , x )= [( − d ) x t − + x t ] l ( D , x ) − bx t x t − − bcx t − > [( − d ) x t − + x t ] bx t − − bx t x t − − bcx t − = bx t − ( − d − c ) ≥ . since c + d = (cid:229) t − i = b + x i + bx t − (cid:229) t − i = x i ≤ . Let G ′ : = ([ t ] , E ∪ F ) , then l ( G ′ , z ) = l ( G , z )) + l ( F , z ) = l ( G , x ) − bx x t − + l ( F , z ) > l ( G , x ) . On theother hand, since z has only t − l ( G ′ , z ) < l ([ t − ] ( ) ) . ⊓⊔ Proof of Lemma 4.4
Let m and t be positive integers satisfying (cid:0) t − r (cid:1) ≤ m ≤ (cid:0) tr (cid:1) − . Let G : = ( V , E ) be an r -graphon vertex set V : = [ n ] with m edges containing a clique of size t − l ( G ) = l r ( m , t − ) . We call such a G anextremal r -graph for m and t −
2. Let x : = ( x , x , . . ., x n ) be an optimal weighting of G . We can assume that x i ≥ x j when i < j since otherwise we can just relabel the vertices of G and obtain another extremal r -graph for m and t − x : = ( x , x , . . ., x n ) satisfying x i ≥ x j when i < j . If G is not left-compressed, then there is anedge whose ancestor is not an edge. Replace all those edges by its available ancestor with the highest hierarchy,then we get a left-compressed r -graph G ′ which contains the clique [ t − ] ( r ) and l ( G ′ , x ) ≥ l ( G , x ) . ⊓⊔ Proof of Lemma 5.3
Let b : = max { i : i ( t − ) t ∈ E } . Since E is left-compressed, then E i = { , . . . , i − , i + , . . ., t } ( ) , for 1 ≤ i ≤ b , and E i \ j = /0 for 1 ≤ i < j ≤ b .Hence, by Remark 2.2(a), we have x = x = · · · = x b . We define a new feasible weighting y for G as follows.Let y i : = x i for i = t − , t , y t − : = x t − + x t and y t : = l ( E t − , x ) = l ( E t , x ) , so l ( G , y ) − l ( G , x ) = x t ( l ( E t − , x ) − x t l ( E t ( t − ) , x )) − x t ( l ( E t , x ) − x t − l ( E ( t − ) t , x )) − x t − x t l ( E ( t − ) t , x )= x t ( l ( E t − , x ) − l ( E t , x )) − x t b (cid:229) i = x i = − bx x t . (28)Since y t = t from E to form a new 3-graph G : = ([ t ] , E ) with | E | : = | E | − | E t | and l ( G , y ) = l ( G , y ) .We will show that if | [ t − ] ( ) \ E | ≥ t − F ⊂ [ t − ] ( ) \ E satisfying l ( F , y ) ≥ bx x t , (29)Then, using (28), (29), the 3-graph G ′ : = ([ t ] , E ′ ) , where E ′ : = E ∪ F , satisfies l ( G ′ , y ) = l ( G , y ) + l ( F , y ) ≥ l ( G , y ) + bx x t = l ( G , x ) . Since y has only t − l ( G ′ ) ≤ l ([ t − ] ( ) ) , and consequently l ( G ) ≤ l ([ t − ] ( ) ) . We must now construct the set of edges F satisfying (29). Applying Remark 2.2(a) by taking i = , j = t − x = x t − + l ( E \ ( t − ) , x ) l ( E ( t − ) , x ) . Let C : = [ t − ] ( ) \ E t − . Then l ( E \ ( t − ) , x ) = x t (cid:229) t − i = b + x i + l ( C , x ) . Applying this and multiplying bx t to theabove equation (note that l ( E ( t − ) , x ) = (cid:229) ti = , i = t − x i ), we have bx x t = bx t − x t + bx t (cid:229) t − i = b + x i (cid:229) ti = , i = t − x i + bx t l ( C , x ) (cid:229) ti = , i = t − x i . Since x ≥ x ≥ · · · ≥ x t , then bx x t ≤ bx t − x t ( + t − ( b + ) t − ) + bx t l ( C , x ) t − . (30)Define a : = ⌈ b | C | t − ⌉ and b : = ⌈ b ( + t − ( b + ) t − ) ⌉ . Note that since b ≤ t −
2. So b ≤ t −
2. Let the set F ⊂ [ t − ] ( ) \ E consist of the a heaviest edges in [ t − ] ( ) \ E containing the vertex t − | [ t − ] ( ) \ E t − | = | C | ≥ a ). Recalling that y t − = x t − + x t we have l ( F , y ) ≥ bx t l ( C , x ) t − + a x t − x t . So using (30) l ( F , y ) − bx x t ≥ x t − x t ( a − b ) . (31)If a > b , then l ( F , y ) − bx t − x t >
0. So defining F : = F satisfies (29).Assume a ≤ b . Suppose that | [ t − ] ( ) \ E | ≥ t −
2. So | [ t − ] ( ) \ E | ≥ t − ≥ b (recall that b ≤ t − F consist of any b − a edges in [ t − ] ( ) \ ( E ∪ F ) and define F : = F ∪ F . Then since l ( F , y ) ≥ ( b − a ) x t − and using (30) l ( F , y ) − bx t − x t = l ( F , y ) − bx t − x t + l ( F , y ) ≥ ( b − a ) x t − − x t − x t ( b − a ) ≥ . This proves Lemma 5.3. ⊓⊔ At this moment, we are not able to extend the arguments in this paper to verify Conjectures 2.1, 2.2, and 2.3 formore general cases. When r ≥
4, the computation is more complex. If there is some technique to overcome thisdifficulty, then the idea used in proving Theorem 3.1 can be used to improve our results much further.
Acknowledgments
We thank two anonymous referees and the editor for helpful and insightful comments. Thisresearch is partially supported by National Natural Science Foundation of China (No. 11271116). n Lagrangians of Hypergraphs Containing Dense Subgraphs 21
References
1. Tur´an, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok. , 436-452 (1941)2. Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Tur´an. Canad. J. Math. ,533-540 (1965)3. Bomze, I.M.: Evolution towards the maximum clique. J. Glob. Optim. , 143-164 (1997)4. Budinich, M.: Exact bounds on the order of the maximum clique of a graph. Discret Appl. Math. , 535-543(2003)5. Busygin, S.: A new trust region technique for the maximum weight clique problem. Discret Appl. Math. ,2080-2096 (2006)6. Gibbons, L.E., Hearn, D. W., Pardalos, P. M., Ramana, M. V.: Continuous characterizations of the maximumclique problem. Math. Oper. Res. , 754-768 (1997)7. Pavan, M., Pelillo M.: Generalizing the Motzkin-Straus theorem to edge-weighted graphs, with applications toimage segmentation. In: Rangarajan A., Figueiredo, Mrio A. T., Zerubia J. (Eds.): Lecture Notes in ComputerScience, vol. , pp. 485-500. Spring, New York (2003)8. Pardalos, P.M., Phillips, A.: A global optimization approach for solving the maximum clique problem. Int. J.Comput. Math. , 209-216 (1990)9. Bul´o,S. R., Torsello, A., Pelillo, M.: A continuous-based approach for partial clique enumeration. In: EscolanoF., Vento M. (Eds.): Lecture Notes in Computer Science, vol. , pp. 61-70. Spring, New York (2007)10. Sidorenko, A. F.: Solution of a problem of Bollob´as on 4-graphs. Mat. Zametki. , 433-455(1987)11. Frankl, P., F¨uredi, Z.: Extremal problems whose solutions are the blow-ups of the small Witt-designs. J.Combin. Theory, Ser. A. , 129-147 (1989)12. Frankl, P., R¨odl, V.: Hypergraphs do not jump. Combinatorica. , 149-159 (1984)13. S´os, V.T., Straus, E.G.: Extremals of functions on graphs with applications to graphs and hypergraphs. J.Combin. Theory, Ser. A. , 246-257 (1982)14. Bul`o, S.R., Pelillo, M.: A continuous characterization of maximal cliques in k -uniform hypergraphs. In:Maniezzo V., Battiti R., Watson, J. P. (Eds.): Lecture Notes in Computer Science, vol. , pp. 220-233. Spring,New York (2008)15. Bul`o, S.R., Pelillo, M.: A generalization of the Motzkin-Straus theorem to hypergraphs. Optim. Lett. , 287-295 (2009)16. Peng, Y., Zhao, C.: A Motzkin-Straus type result for 3-uniform hypergraphs. Graphs Comb. , 681-694(2013)17. Talbot, J.M.: Lagrangians of hypergraphs. Comb. Probab. Comput. , 199-216 (2002)18. Mubayi, D.: A hypergraph extension of Tur´an’s theorem. J. Combin. Theory, Ser. B. , 122-134 (2006)19. Peng, Y., Tang, Q., Zhao, C.: On Lagrangians of rr