On hypergraph Lagrangians

TL;DR

This paper investigates the maximum Lagrangian of r-uniform hypergraphs with a given number of edges, establishing new bounds and connections to clique numbers, and confirming conjectures for specific cases.

Contribution

It extends the understanding of hypergraph Lagrangians by linking them to clique numbers and providing new bounds for hypergraphs with certain edge and vertex counts.

Findings

Confirmed the conjecture for specific ranges of m and r.

Established bounds for the Lagrangian based on clique number and colex ordering.

Connected the maximum Lagrangian problem to combinatorial parameters of hypergraphs.

Abstract

It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in \cite{FF}. Motzkin and Straus' theorem confirms this conjecture when $r=2$. For $r=3$, it is shown by Talbot in \cite{T} that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $r$-uniform hypergraphs. As an implication of this connection, we prove that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform graphs with $t$ vertices and $m$ edges satisfying ${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$ for $r\geq 4.$