A Note on the Lagrangian of Linear 3-Uniform Hypergraphs

Abstract

Lots of symmetric properties are well-explored and analyzed in extremal graph theory, such as the well-known symmetrization operation in the Turán problem and the high symmetric in the extremal graphs. This paper is devoted to studying the Lagrangian of hypergraphs, which connects to a very symmetric function—the Lagrangian function. Given an r-uniform hypergraph F, the Lagrangian density πλ(F) is the limit supremum of r!λ(G) over all F-free G, where λ(G) is the Lagrangian of G. An r-uniform hypergraph F is called λ-perfect if πλ(F) equals r!λ(Kv(F)−1r). Yan and Peng conjectured that: for integer r≥3, there exists n0(r) such that if G and H are two λ-perfect r-graphs with |V(G)| and |V(H)| no less than n0(r), then the disjoint union of G and H is λ-perfect. Let St denote a 3-uniform hypergraph with t edges {e1,⋯,et} satisfying that ei∩ej={v} for all 1≤i<j≤t. In this paper, we show that the conjecture holds for G=S2 and H=St for all t≥62. Moreover, our result yields a class of Turán densities of 3-uniform hypergraphs. In our proof, we use some new techniques to study Lagrangian density problems; using a result of Sidorenko to find subgraphs, and a result of Talbot to upper bound the Lagrangian of a hypergraph.