[Paper Review] Perfect matchings in random sparsifications of Dirac hypergraphs

Dong Yeap Kang, Tom Kelly+3 more|arXiv (Cornell University)|Nov 2, 2022

Limits and Structures in Graph Theory4 citations

TL;DR

This paper establishes robust Dirac-type theorems for perfect matchings in random sparsifications of $k$-uniform hypergraphs. For $n \in k\mathbb{N}$ and $p = \Omega(n^{-k+1}\log n)$, if a hypergraph $\mathcal{H}$ satisfies the minimum $(k-1)$-degree condition $\delta_{k-1}(\mathcal{H}) \geq m_{k-1}(k,n)$, then its $p$-random subhypergraph $\mathcal{H}_p$ a.a.s. contains a perfect matching. The result extends to lower-degree conditions with an additive $\gamma\binom{n-d}{k-d}$ term, and the number of perfect matchings is shown to be at least $\exp((1-1/k)n\log n - \Theta(n))$, optimal up to a $\exp(\Theta(n))$ factor.

ABSTRACT

For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $δ_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = Ω(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching. Moreover, for every fixed integer $d < k$ and $γ> 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_d(\mathcal H) \geq m_{d}(k,n) + γ\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - Θ(n))$ many perfect matchings, which is best possible up to an $\exp(Θ(n))$ factor.

Motivation & Objective

• To resolve a robust version of Shamir’s problem by establishing conditions under which random sparsifications of $k$-uniform hypergraphs contain perfect matchings.
• To extend Dirac-type theorems to random subhypergraphs by proving that minimum $d$-degree conditions with an additive $\gamma\binom{n-d}{k-d}$ term suffice for a.a.s. perfect matchings.
• To determine the asymptotic number of perfect matchings in such hypergraphs, showing it is at least $\exp((1-1/k)n\log n - \Theta(n))$, matching the best possible bound up to a $\exp(\Theta(n))$ factor.
• To unify extremal and probabilistic hypergraph theory by proving that random sparsifications of hypergraphs satisfying Dirac-type minimum degree conditions retain perfect matchings with high probability.

Proposed method

• Use of the absorbing method to construct a nearly perfect matching and then absorb the remaining vertices into a perfect matching.
• Application of the spreadness and typicality framework to control the structure of random subhypergraphs via pseudorandomness properties.
• Employment of Chernoff bounds and concentration inequalities to show that random subsets of vertices preserve minimum degree conditions with high probability.
• Use of the $\alpha$-typicality condition and degree distribution analysis to compare the random subhypergraph to a canonical hypergraph $\mathcal{K}_{r^*}$.
• Construction of a random partitioning of the vertex set into $U_\ell$ with controlled size and degree distribution, ensuring typicality with high probability.
• Establishment of a hierarchy of degree conditions across multiple levels of hypergraph structure, using iterative absorption and probabilistic embedding techniques.

Experimental results

Research questions

• RQ1 Under what minimum $d$-degree conditions on a $k$-uniform hypergraph does its $p$-random subhypergraph a.a.s. contain a perfect matching?
• RQ2 Can the threshold for perfect matchings in random $k$-uniform hypergraphs be robustly preserved under sparsification when the original hypergraph satisfies a Dirac-type minimum codegree condition?
• RQ3 What is the minimum number of perfect matchings that exist in a $k$-uniform hypergraph satisfying the $m_{k-1}(k,n)$ codegree threshold, and how does this number scale with $n$?
• RQ4 How does the addition of an $\gamma\binom{n-d}{k-d}$ term to the minimum $d$-degree condition affect the existence of perfect matchings in random subhypergraphs?

Key findings

• For $n \in k\mathbb{N}$ and $p = \Omega(n^{-k+1}\log n)$, if $\delta_{k-1}(\mathcal{H}) \geq m_{k-1}(k,n)$, then $\mathcal{H}_p$ contains a perfect matching with high probability.
• For any fixed $d < k$ and $\gamma > 0$, if $\delta_d(\mathcal{H}) \geq m_d(k,n) + \gamma\binom{n-d}{k-d}$, then $\mathcal{H}_p$ a.a.s. contains a perfect matching.
• The number of perfect matchings in such hypergraphs is at least $\exp((1-1/k)n\log n - \Theta(n))$, which is optimal up to a $\exp(\Theta(n))$ factor.
• The results provide robust versions of Dirac-type theorems for hypergraphs, extending the solution to Shamir’s problem to the setting of random sparsifications.
• The proof relies on a combination of the absorbing method, pseudorandomness, and concentration inequalities to control the structure of random subhypergraphs.