摘要
The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turin classical result on the Turan density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t - 1, then A（G） = A（[t- 1]（3）） provided （t31） ≤ m ≤ （3^t1） ＋ （2^rt-2）. They also conjectured： If G is an r-uniform graph with m edges not containing a clique of order t - 1, then A（G） 〈 A（[t - 1]（r）） provided （r^t-1） ≤ m ≤ （r^t-1） ＋ （r-1^t-2）. It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m = （3^t-1） ＋ （2^t-2）. Regarding this conjecture, we show： If G is a left-compressed 3-uniform graph on the vertex set It] with m edges and lit - 1]（3） ／ E（G）｜=- p, then A（G） 〈 A（[t - 1]（3）） provided m = （3^t-1） ＋ （2^t-2） and t ≥ 17p/2 ＋ 11.
The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turin classical result on the Turan density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t - 1, then A（G） = A（[t- 1]（3）） provided （t31） ≤ m ≤ （3^t1） ＋ （2^rt-2）. They also conjectured： If G is an r-uniform graph with m edges not containing a clique of order t - 1, then A（G） 〈 A（[t - 1]（r）） provided （r^t-1） ≤ m ≤ （r^t-1） ＋ （r-1^t-2）. It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m = （3^t-1） ＋ （2^t-2）. Regarding this conjecture, we show： If G is a left-compressed 3-uniform graph on the vertex set It] with m edges and lit - 1]（3） ／ E（G）｜=- p, then A（G） 〈 A（[t - 1]（3）） provided m = （3^t-1） ＋ （2^t-2） and t ≥ 17p/2 ＋ 11.
基金
Supported in part by National Natural Science Foundation of China(Grant No.11271116)
