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 (₃ ^t-1) ≤ m ≤ (₃ ^t-1) + (₂ ^t-2). 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 (_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 [t] with m edges and |[t-1]⁽³⁾E(G)| = p, then λ(G) < λ([t-1]⁽³⁾) provided m = (³ _t-1) + (² _t-2) and t ≥ 17p/2 + 11.