[Paper Review] Lagrangian densities of hypergraph cycles

Yuejian Peng, Zilong Yan|arXiv (Cornell University)|Oct 31, 2018

Limits and Structures in Graph Theory10 references4 citations

TL;DR

This paper establishes the Lagrangian density of the 3-uniform hypergraph $F_5 = \{123,124,345\}$, proving it is the first known non-perfect 3-uniform hypergraph, and confirms the perfection of the 3-uniform linear cycle $C_3^3$. It further shows that $C_3^3$-free 3-graphs achieve maximum Lagrangian only when containing $K_5^3$, and extends results to $r$-uniform hypergraphs using transference techniques.

ABSTRACT

The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $π_λ(H)\ge r!λ{(K_{t-1}^r)}$. We say that an $r$-unform hypergraph $H$ with $t$ vertices is perfect if $π_λ(H)= r!λ{(K_{t-1}^r)}$. A theorem of Motzkin-Straus implies that all $2$-uniform graphs are perfect. It is interesting to explore what kind of hypergraphs are perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For $r\ge 3$, there exists $n$ such that a linear $r$-unofrm hypergraph with at least $n$ vertices is perfect. (2) For $r\ge 3$, there exists $n$ such that if $G, H$ are perfect $r$-uniform hypergraphs with at least $n$ vertices, then $G\bigsqcup H$ is perfect. Regarding this conjecture, we obtain a partial result: Let $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$. (An earlier result of Sidorenko states that $S_{2,t}$ is perfect \cite{Sidorenko-89}.) Let $H$ be a perfect $3$-graph with $s$ vertices. Then $F=S_{2,t}\bigsqcup H$ is perfect if $s\geq 3$ and $t\geq 3$.

Motivation & Objective

• To determine the Lagrangian density of $F_5 = \{123,124,345\}$, a 3-uniform hypergraph, and resolve one of three unsolved cases for 3-uniform hypergraphs on three edges.
• To investigate the perfection of $r$-uniform hypergraphs, particularly linear hypergraphs, by extending the conjecture that large linear $r$-graphs are perfect.
• To establish that the 3-uniform linear cycle $C_3^3 = \{123,345,561\}$ is perfect, and that only hypergraphs containing $K_5^3$ achieve the maximum Lagrangian among $C_3^3$-free 3-graphs.
• To extend results on Lagrangian densities to $r$-uniform hypergraphs, particularly for linear cycles of length $t$, using transference techniques.
• To resolve the long-standing open problem of Lagrangian densities for hypergraph cycles, providing the first non-perfect example in 3-uniform hypergraphs.

Proposed method

• Applies the Lagrangian method to $F_5$, using optimization over feasible weight vectors $\vec{x} \in \Delta$ to compute $\lambda(G)$ for $F_5$-free 3-graphs.
• Employs a transference technique of Pikhurko to relate Turán densities of hypergraph extensions to Lagrangian densities.
• Uses structural analysis of $F_5$-free 3-graphs $G$ by considering vertex extensions beyond $K_{2t-2}^{3-}$, analyzing neighborhood conditions to avoid $F_5$-copies.
• Applies combinatorial facts (e.g., Fact 5.8, Fact 5.17) to show that if certain good pair conditions hold, then $\lambda(G) < \lambda(K_{2t-1}^3)$, implying non-optimality.
• Analyzes cases based on edge configurations involving new vertices $x$ and $a$, using path constructions (e.g., $P_{t-3}$) to detect forbidden cycles.
• Proves that $C_3^3$-free 3-graphs achieving maximum Lagrangian must contain $K_5^3$, using neighborhood and edge-avoidance arguments to rule out alternative configurations.

Experimental results

Research questions

• RQ1 What is the Lagrangian density of the 3-uniform hypergraph $F_5 = \{123,124,345\}$, and is it perfect?
• RQ2 Is the 3-uniform linear cycle $C_3^3 = \{123,345,561\}$ perfect, and which $C_3^3$-free 3-graphs achieve the maximum Lagrangian?
• RQ3 Can the perfection property be extended to $r$-uniform hypergraphs, particularly for linear cycles of length $t$?
• RQ4 Do the results on $F_5$ and $C_3^3$ extend to $r$-uniform hypergraphs via transference techniques?
• RQ5 Under what conditions is the disjoint union of two perfect $r$-graphs also perfect?

Key findings

• The Lagrangian density of $F_5 = \{123,124,345\}$ is $\lambda(K_5^3)$, and $F_5$ is the first known non-perfect 3-uniform hypergraph.
• $C_3^3 = \{123,345,561\}$ is proven to be perfect, and among all $C_3^3$-free 3-graphs, only those containing $K_5^3$ achieve $\lambda(G) = \lambda(K_5^3)$.
• For $t \geq 3$, the disjoint union $S_{2,t} \sqcup H$ is perfect if $H$ is a perfect 3-graph on $s \geq 3$ vertices, extending Sidorenko’s earlier result.
• The 3-uniform linear cycle $C_t^3$ of length $t$ is perfect for $t \geq 3$, and its Lagrangian density is $\lambda(K_{2t-1}^3)$.
• The Turán density of extensions of $F_5$, $C_3^3$, and $S_{2,t}$ can be derived via Pikhurko’s transference technique.
• If $G$ is a $C_3^3$-free 3-graph and $\lambda(G) = \lambda(K_5^3)$, then $G$ must contain $K_5^3$ as a subgraph, confirming extremality.