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On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges |
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| Authors: | Yanping Sun Qingsong Tang Cheng Zhao Yuejian Peng |
| Institution: | 1. College of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China 2. College of Sciences, Northeastern University, Shenyang, 110819, People’s Republic of China 3. Mathematics School, Institute of Jilin University, Changchun, 130012, People’s Republic of China 4. Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA 5. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China |
| Abstract: | The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the \({r}\) -graph with \(m\) edges formed by taking the first \(\textit{m}\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({r}\) has the largest Graph-Lagrangian of all \(r\) -graphs with \(m\) edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed \(3\) -graphs with \(m\) edges is at most the Graph-Lagrangian of the \(\mathrm 3 \) -graph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({3}\) . |
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