Lagrangian-Perfect Hypergraphs

Hypergraph Lagrangian function has been a helpful tool in several celebrated results in extremal combinatorics. Let G be an r-uniform graph on [n] and let \({\textbf{x}}=(x_1,\ldots ,x_n) \in [0,\infty )^n.\) The graph Lagrangian function is defined to be \(\lambda (G,{\textbf{x}})=\sum _{e \in E(G)}\prod _{i\in e}x_{i}.\) The graph Lagrangian is defined as \(\lambda (G)=\max \{\lambda (G, {\textbf{x}}): {\textbf{x}} \in \Delta \},\) where \(\Delta =\{{\textbf{x}}=(x_1,x_2,\ldots ,x_n) \in [0, 1]^{n}: x_1+x_2+\dots +x_n =1 \}.\) The Lagrangian density \(\pi _{\lambda }(F)\) of an r-graph F is defined to be \(\pi _{\lambda }(F)=\sup \{r! \lambda (G): G \text { does not contain }F \}.\) Sidorenko (Combinatorica 9:207–215, 1989) showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turán density of the extension of F. Therefore, determining the Lagrangian density of a hypergraph will add a result to the very few known results on Turán densities of hypergraphs. For an r-uniform graph H with t vertices, \(\pi _{\lambda }(H)\ge r!\lambda {(K_{t-1}^r)}\) since \(K_{t-1}^r\) (the complete r-uniform graph with \(t-1\) vertices) does not contain a copy of H. We say that an r-uniform hypergraph H with t vertices is \(\lambda \)-perfect if the equality \(\pi _{\lambda }(H)= r!\lambda {(K_{t-1}^r)}\) holds. A fundamental theorem of Motzkin and Straus implies that all 2-uniform graphs are \(\lambda \)-perfect. It is interesting to understand the \(\lambda \)-perfect property for \(r\ge 3.\) Our first result is to show that the disjoint union of a \(\lambda \)-perfect 3-graph and \(S_{2,t}=\{123,124,125,126,\ldots ,12(t+2)\}\) is \(\lambda \)-perfect, this result implies several previous results: Taking H to be the 3-graph spanned by one edge and \(t=1,\) we obtain the result by Hefetz and Keevash (J Comb Theory Ser A 120:2020–2038, 2013) that a 3-uniform matching of size 2 is \(\lambda \)-perfect. Doing it repeatedly, we obtain the result in Jiang et al. (Eur J Comb 73:20–36, 2018) that any 3-uniform matching is \(\lambda \)-perfect. Taking H to be the 3-uniform linear path of length 2 or 3 and \(t=1\) repeatedly, we obtain the results in Hu et al. (J Comb Des 28:207–223, 2020). Earlier results indicate that \(K_4^{3-}=\{123, 124, 134\}\) and \(F_5=\{123, 124, 345\}\) are not \(\lambda \)-perfect, we show that the disjoint union of \(K_4^{3-}\) (or \(F_5\)) and \(S_{2,t}\) are \(\lambda \)-perfect. Furthermore, we show the disjoint union of a 3-uniform hypergraph H and \(S_{2,t}\) is \(\lambda \)-perfect if t is large. We also give an irrational Lagrangian density of a family of four 3-uniform hypergraphs.