On Graph-Lagrangians and Clique Numbers of 3-Uniform Hypergraphs

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 Turán classical result on the Turán 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 λ（G）=λ([t-1]（3）) provided （t-13）≤m≤（t-13）+t-22.They also conjectured:If G is an r-uniform graph with m edges not containing a clique of order t-1,then λ（G）<λ([t-1]（r）) provided （t-1r）≤ m ≤（t-1r）+（t-2r-1）.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=t-13+t-22.Regarding this conjecture,we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t-1]（3）\E（G）|=p,then λ（G）<λ([t-1]（3）) provided m=（t-13）+（t-22） and t≥17p/2+11.