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arXiv · 1312.4135

An extension of Motzkin-Straus Theorem to non-uniform hypergraphs and its applications

Yuejian Peng·Hao Peng·Qingsong Tang·Cheng Zhao
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Abstract

In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Tur\'an's theorem using the connection. The connection of Lagrangians and Tur\'{a}n densities can be also used to prove the fundamental theorem of Erd\"{o}s-Stone-Simonovits on Tur\'{a}n densities of graphs. Very recently, the study of Tur\'{a}n densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Tur\'{a}n densities of non-uniform hypergraphs. We first give a definition of the Lagrangian of a non-uniform hypergraph, then give an extension of Motzkin-Straus theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices. Applying it, we give an extension of Erd\"{o}s-Stone-Simonovits theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices.

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Yuejian Peng, Hao Peng, Qingsong Tang, Cheng Zhao. 2013-12-15. An extension of Motzkin-Straus Theorem to non-uniform hypergraphs and its applications. https://arxiv.org/abs/1312.4135

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