Abstract
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 \({\mathbb N}^{(r)}\) 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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Acknowledgments
We thank two anonymous referees for helpful comments. This research is partially supported by National Natural Science Foundation of China (No. 11271116 and No. 61304021) and Chinese Universities Scientific Fund (No. N140504004).
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Tang, Q., Peng, Y., Zhang, X. et al. Connection between the clique number and the Lagrangian of 3-uniform hypergraphs. Optim Lett 10, 685–697 (2016). https://doi.org/10.1007/s11590-015-0907-2
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DOI: https://doi.org/10.1007/s11590-015-0907-2