On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

Abstract

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 $\left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$. 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 $\left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ {r - 1} \end{array}} \right)$. 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 $m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$. 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 $m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)$ and t ≥ 17p/2 + 11.