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On the Keevash-Knox-Mycroft Conjecture

Luyining Gan, Jie Han

TL;DR

This work resolves a central conjecture on perfect matchings in dense $k$-graphs by reducing the decision problem ${\bf PM}(k,\ell,\delta)$ to a threshold problem for perfect fractional matchings. By developing a refined partition lemma, a lattice-based absorption framework, and a polynomial-time algorithm, the authors prove the Keevash–Knox–Mycroft conjecture for all $\ell\ge 0.4k$ and moreover provide an algorithm that outputs a perfect matching when one exists. The core technique separates robust from non-robust edge contributions and localizes divisibility barriers to a finite, testable structure via index/robust vectors and coset solubility. These results bridge extremal hypergraph matching theory and computational tractability, yielding practical polynomial-time procedures for finding perfect matchings under favorable minimum $\ell$-degree conditions. The findings also connect to longstanding conjectures on $c^*_{k,\ell}$ and divisibility constructions, advancing both the theory and its algorithmic implications.

Abstract

Given $1\le \ell <k$ and $δ\ge0$, let $\textbf{PM}(k,\ell,δ)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $δ\binom{n-\ell}{k-\ell}$. For $k\ge 3$, $\textbf{PM}(k,\ell,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $\textbf{PM}(k, \ell, δ)$ is in P for every $δ> 1-(1-1/k)^{k-\ell}$ and verified the case $\ell=k-1$. In this paper we show that this problem can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.

On the Keevash-Knox-Mycroft Conjecture

TL;DR

This work resolves a central conjecture on perfect matchings in dense -graphs by reducing the decision problem to a threshold problem for perfect fractional matchings. By developing a refined partition lemma, a lattice-based absorption framework, and a polynomial-time algorithm, the authors prove the Keevash–Knox–Mycroft conjecture for all and moreover provide an algorithm that outputs a perfect matching when one exists. The core technique separates robust from non-robust edge contributions and localizes divisibility barriers to a finite, testable structure via index/robust vectors and coset solubility. These results bridge extremal hypergraph matching theory and computational tractability, yielding practical polynomial-time procedures for finding perfect matchings under favorable minimum -degree conditions. The findings also connect to longstanding conjectures on and divisibility constructions, advancing both the theory and its algorithmic implications.

Abstract

Given and , let be the decision problem for the existence of perfect matchings in -vertex -uniform hypergraphs with minimum -degree at least . For , was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that is in P for every and verified the case . In this paper we show that this problem can be reduced to the study of the minimum -degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for . Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.
Paper Structure (20 sections, 21 theorems, 48 equations)

This paper contains 20 sections, 21 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Minimum degree conditions and divisibility barriers
  3. Related work
  4. A partition lemma and a structural theorem
  5. Reachability
  6. Index vector and robust vector
  7. Lattices, solubility and the structural theorem
  8. Highlights of the proof: a comparison with the Han--Treglown proof
  9. Proof of Theorem \ref{['thm_main']}
  10. Useful tools
  11. Almost perfect matchings
  12. Build a partition
  13. Proof of Lemma \ref{['lem:partition']}
  14. Absorption lemmas
  15. Absorption lemmas for Theorem \ref{['thm:alm_mat']}
  16. ...and 5 more sections

Key Result

Theorem 1.3

Suppose $k,\ell \in \mathbb N$ such that $1\le \ell \leq k-1$. Then for any $\delta \in (c^*_{k, \ell},1]$, ${\bf PM}(k, \ell, \delta)$ is in $P$. That is, for any $\delta \in (c^*_{k, \ell},1]$, there exists a constant $c=c(k)$ such that there is an algorithm with running time $O(n^c)$ which given

Theorems & Definitions (33)

  • Conjecture 1.2: Keevash, Knox and Mycroft KKM2015
  • Theorem 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Conjecture 1.6: Hàn--Person-Schacht, HPS
  • Definition 2.1: $\mu$-robust vectors
  • Lemma 2.2
  • Theorem 2.3: Structural Theorem
  • Proposition 5.1
  • Theorem 5.2
  • ...and 23 more