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Connection Between Continuous Optimization and Turán Densities of Non-uniform Hypergraphs

Connection Between Continuous Optimization and Turán Densities of Non-uniform Hypergraphs A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques. Applying it, they gave a new proof of Turán’s theorem. This aroused the interests in studying the connection between continuous optimization and extremal problems in combinatorics. In 2009, S. Rota Bulò and M. Pelillo extended the result of Motzkin-Straus to r-uniform hypergraphs. Recently, Johnston and Lu initiated the study of the Turán density of a non-uniform hypergraph. Polynomial optimization problems related to several types of non-uniform hypergraphs and its applications on Turán densities have also been studied. In this paper, we obtain a Motzkin-Straus type of results for all non-uniform hypergraphs. Applying it, we give an upper bound of the Turán density of a complete non-uniform hypergraph. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Connection Between Continuous Optimization and Turán Densities of Non-uniform Hypergraphs

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Publisher
Springer Journals
Copyright
Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2021
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-021-1048-2
Publisher site
See Article on Publisher Site

Abstract

A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques. Applying it, they gave a new proof of Turán’s theorem. This aroused the interests in studying the connection between continuous optimization and extremal problems in combinatorics. In 2009, S. Rota Bulò and M. Pelillo extended the result of Motzkin-Straus to r-uniform hypergraphs. Recently, Johnston and Lu initiated the study of the Turán density of a non-uniform hypergraph. Polynomial optimization problems related to several types of non-uniform hypergraphs and its applications on Turán densities have also been studied. In this paper, we obtain a Motzkin-Straus type of results for all non-uniform hypergraphs. Applying it, we give an upper bound of the Turán density of a complete non-uniform hypergraph.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 1, 2021

Keywords: hypergraph; maximum clique; polynomial optimization; 05C65; 05D05

References