The Lagrangian density of the disjoint union of a $3$-uniform tight path and a matching and the Turán number of its extension

Given a positive integer $n$ and an $r$-uniform hypergraph $F$, the Turán number $ex(n, F)$ is the maximum number of edges in an $F$-free $r$-uniform hypergraph on $n$ vertices. The Turán density of $F$ is defined as $\pi (F) = \lim_{n \to \infty} ex (n, F) / \binom{n}{r}$. The Lagrangian density of an $r$-uniform graph $F$ is $\pi_\lambda (F) = \sup{\lbrace r \lambda (G) : G \: \textrm{is} \: F-\textrm{free} \rbrace}$, where $ \lambda (G)$ is the Lagrangian of $G$. In 1989, Sidorenko [20] showed that the Lagrangian density of a hypergraph $F$ is the same as the Turán density of its extension. For an $r$-uniform graph $F$ on $t$ vertices, it is clear that $ \pi_\lambda (F) \geq r ! \lambda (K^r_{t-1})$, where $ K^r_{t-1}$ is the complete $r$-uniform graph on $t-1$ vertices. We say that an $r$-uniform hypergraph $F$ on $t$ vertices is $\lambda$-perfect if $ \pi \lambda (F) = r ! \lambda (K^r_{t-1})$. A result of Motzkin and Straus implies that all graphs are $\lambda$-perfect. A conjecture proposed in [23] states that for $r \geq 3$, there exists an integer $n$ such that if $F$ and $H$ are $\lambda$-perfect $r$-uniform graphs on at least $n$ vertices, then the disjoint union of $F$ and $H$ is $\lambda$-perfect. The conjecture has been verified in [23] for a $3$-uniform tight star $T_t = \lbrace 123, 124, \dotsc , 12(t + 2) \rbrace$ and a $\lambda$-perfect $3$-uniform graph for $t \geq 3$ (Sidorenko [20] showed that $T_t$ is $\lambda$-perfect). The case $t = 2$ remains unsolved. In this paper, we shall show that the disjoint union of $T_2 \cong \lbrace 123, 234 \rbrace$ and a $3$-uniform matching is $\lambda$-perfect (Jiang–Peng–Wu [9] showed that a $3$-uniform matching is $\lambda$-perfect). Moreover, using a stability argument of Pikhurko [16], together with a transference technique between the Lagrangian density of an $r$-uniform graph and the Turán density of its extension, we also obtain the Turán numbers of their extensions.

Keywords

hypergraph Lagrangian, Lagrangian density, Turán number

2010 Mathematics Subject Classification

05C35, 05C65

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Pingge Chen is supported by National Natural Science Foundation of China (No. 12101221), National Natural Science Foundation of Hunan Province, China (No. 2021JJ30208).

Yuejian Peng is supported in part by National Natural Science Foundation of China (No. 11931002).

Received 31 May 2021

Received revised 6 March 2022

Accepted 18 September 2022

Published 29 March 2023