[Paper Review] Counting Hypergraphs with Large Girth

Sam Spiro, Jacques Verstraëte|arXiv (Cornell University)|Oct 4, 2020

Limits and Structures in Graph Theory22 references4 citations

TL;DR

This paper extends the container method to count $r$-uniform hypergraphs with large girth by relating the number of such hypergraphs to the number of large-girth graphs. It proves that the number of $n$-vertex $r$-graphs with $m$ edges and girth greater than $\ell$ is bounded by $\mathrm{N}_m^2(n,\ell)^{r-1+\lambda}$, where $\lambda = \lceil(r-2)/(\ell-2)\rceil$, which is tight up to a $1+o(1)$ term in the exponent when $\ell-2$ divides $r-2$. This provides a strong quantitative link between extremal graph and hypergraph theory for sparse, high-girth structures.

ABSTRACT

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The {\em girth} of a hypergraph $H$ is the minimum $\ell$ such that for some $F \subseteq H$, there exists a bijection $ϕ: E(C_\ell) o E(F)$ with $e\subseteq ϕ(e)$ for all $e\in E(C_\ell)$. Letting $N_m^r(n,\ell)$ denote the number of $n$-vertex $r$-uniform hypergraphs with $m$ edges and girth larger than $\ell$ and defining $λ= \lceil (r - 2)/(\ell - 2) floor$, we show \[ N_m^r(n,\ell) \leq N_m^2(n,\ell)^{r - 1 + λ}\] which is tight when $\ell - 2 $ divides $r - 2$ up to a $1 + o(1)$ term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than $\ell$ in random $r$-uniform hypergraphs.

Motivation & Objective

• To extend the container method from graphs to $r$-uniform hypergraphs for counting structures with large girth.
• To establish tight upper bounds on the number of $n$-vertex $r$-graphs with $m$ edges and girth greater than $\ell$
• To address the extremal problem for subgraphs of girth more than $\ell$ in random $r$-uniform hypergraphs.
• To determine the asymptotic behavior of the extremal function $\mathrm{ex}(H_{n,p}^r, \mathcal{C}_{[\ell]}^r)$ in random hypergraphs.

Proposed method

• Uses the container method to bound the number of $r$-graphs with $m$ edges and girth $> \ell$, generalizing results from graphs.
• Defines $\lambda = \lceil(r-2)/(\ell-2)\rceil$ and proves $\mathrm{N}_m^r(n,\ell) \leq \mathrm{N}_m^2(n,\ell)^{r-1+\lambda}$, relating hypergraph counts to graph counts.
• Applies probabilistic and extremal techniques to analyze the number of Berge cycles in random hypergraphs.
• Employs Markov’s inequality and Chernoff bounds to establish concentration of edge and cycle counts in random hypergraphs.
• Uses double counting and extremal bounds on Berge cycle appearances to derive tightness conditions.
• Introduces a generalization to Berge-$F$ hypergraphs for forests $F - v$ and proposes a conjecture on optimal exponents.

Experimental results

Research questions

• RQ1 How can the container method be adapted to count $r$-uniform hypergraphs with large girth, given the lack of effective bounds for $\mathrm{N}_m^r(n,\ell)$?
• RQ2 What is the tightest possible exponent $\beta$ such that $\mathrm{N}_m^r(n,\mathcal{C}_\ell^r) \leq 2^{cm} \cdot \mathrm{N}_{[m]}^2(n,C_\ell)^\beta$?
• RQ3 Does the extremal number of edges in a random $r$-uniform hypergraph with girth more than $\ell$ follow a two-phase threshold behavior as conjectured?
• RQ4 Can the exponent $r!/2$ in the bound for Berge-$F$ hypergraphs be improved, especially when $F$ is a theta graph?
• RQ5 Under what conditions does $p^m \cdot \mathrm{N}_m^r(n,\ell) \to \infty$ imply that $H_{n,p}^r$ a.a.s. contains a subgraph of girth $> \ell$ with $m$ edges?

Key findings

• The number of $n$-vertex $r$-graphs with $m$ edges and girth greater than $\ell$ is bounded by $\mathrm{N}_m^2(n,\ell)^{r-1+\lambda}$, where $\lambda = \lceil(r-2)/(\ell-2)\rceil$, providing a strong quantitative link to graph counts.
• This bound is tight up to a $1+o(1)$ term in the exponent when $\ell-2$ divides $r-2$, confirming optimality in key cases.
• For random $r$-uniform hypergraphs $H_{n,p}^r$, the extremal number of edges in a subgraph of girth $> \ell$ is shown to be at least $p^{1/(2r-3)}n^{2-o(1)}$ a.a.s., establishing a lower bound.
• The method fails for $\ell \geq 4$ in the current form due to structural issues with Berge 4-cycles in the auxiliary hypergraph $H''$, limiting applicability.
• A generalization to Berge-$F$ hypergraphs is proven when $F - v$ is a forest, yielding a bound with exponent $r!/2$, though this is likely not optimal.
• The paper proposes Conjecture III on a two-phase extremal threshold in random hypergraphs, with a candidate exponent $\gamma = r-1 + (r-2)/(ε-2)$, and poses Problem II on verifying $\gamma(3,4) = 5/2$.