A Motzkin–Straus Type Result for 3-Uniform Hypergraphs

Peng, Yuejian; Zhao, Cheng

doi:10.1007/s00373-012-1135-5

Cited by 20 publications

(32 citation statements)

References 16 publications

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“…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 optimization problem of a class of homogeneous multilinear functions over the standard simplex of the Euclidean space.…”

Section: Introductionsupporting

confidence: 81%

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On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

Tang

Peng

Zhang

et al. 2013

J Optim Theory Appl

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Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002).

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Section: Introductionsupporting

confidence: 81%

“…Then λ (G) = λ ([t − 1] (r) ). [16], Conjecture 1.4) Let m and t be positive integers satisfying t−1 r ≤ m ≤ t−1 r + t−2 r−1 . Let G be an r-graph with m edges and contain no clique of order t − 1.…”

Section: Theorem 21 (See [2] Theorem 1) If G Is a 2-graph With N Vementioning

confidence: 99%

On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

Tang

Peng

Zhang

et al. 2013

J Optim Theory Appl

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“…Lemma 1 [12] For any integers m, t, and r satisfying t−1 In [7], we showed that Conjecture 2 holds when r = 3 as in the following Theorem.…”

Section: Introductionmentioning

confidence: 91%

“…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 its cliques number for hypergraphs when the number of edges is in certain ranges. In [7], it is conjectured that the following Motzkin and Straus type results are true for hypergraphs. The upper bound t−1 r + t−2 r−1 in this conjecture is the best possible.…”

Section: Introductionmentioning

confidence: 98%

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

et al. 2015

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There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533-540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. Frankl and Füredi conjectured that the r -uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of N (r ) has the largest Lagrangian of all r -uniform hypergraphs with m edges. For r = 2, Motzkin and Straus' theorem confirms this conjecture. For r = 3, it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi's conjecture holds for bigger ranges of m when r = 3. We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.

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“…[8]) Let t, m and r ≥ 3 be positive integers satisfying t−+ t−2 r−1 . Let G be an r-graph with m edges and G contain a clique of order t − 1.…”

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confidence: 99%

Dense 3-uniform hypergraphs containing a large clique

Peng

2017

Sci. China Math.

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An r-uniform graph G is dense if and only if every proper subgraph G ′ of G satisfies λ(G ′ ) < λ(G), where λ(G) is the Lagrangian of a hypergraph G. In 1980's, Sidorenko showed that π(F ), the Turán density of an r-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F -hom-free r-uniform hypergraphs. This connection has been applied in estimating Turán density of hypergraphs. When r = 2, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when r ≥ 3, it becomes much harder to estimate the Lagrangians of r-uniform hypergraphs and to characterize the structure of all dense r-uniform graphs. The main goal of this note is to give some sufficient conditions for 3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [t] and m edges containing [t − 1] (3) , then G is dense if and only if m ≥ t−1 3 + t−2 2 + 1. We also give sufficient condition condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.

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A Motzkin–Straus Type Result for 3-Uniform Hypergraphs

Cited by 20 publications

References 16 publications

On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

On Graph-Lagrangians of Hypergraphs Containing Dense Subgraphs

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

Dense 3-uniform hypergraphs containing a large clique

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