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Advancing the Rödl Nibble: New bounds on matchings and the list chromatic index of hypergraphs

Stephen Gould, Tom Kelly

TL;DR

This work advances the Rödl nibble framework for hypergraph matchings by exhaustively exploiting the full codegree sequence up to natural bottlenecks, enabling near-perfect coverings with subpolynomial errors. It introduces a robust pseudorandom matching paradigm via a family of weight functions, yielding applications to list colorings and to a spectrum of combinatorial structures including Latin squares, Steiner systems, and simplicial complexes. The core methodology combines a two-stage Nibble/Chomp process with concentration tools (including Linear Talagrand with exceptional outcomes and Kim–Vu polynomial concentration) and absorption-inspired reservoir techniques to convert almost-matchings into near-perfect or perfect ones in complex, multipartite settings. The results yield improved bounds on the list chromatic index, as well as near-optimal leftovers for diverse combinatorial objects, highlighting the practical impact of nibble-based approaches in design theory, Latin squares, and higher-dimensional combinatorial constructions.

Abstract

Let $H$ be a $(k+1)$-uniform hypergraph which is nearly $D$-regular, such that any set of $i$ vertices is contained in at most $D_i$ edges of $H$ for each $i = 2, 3, \dots, k+1$. Influential results of Pippenger and of Frankl and Rödl show that the \textit{Rödl Nibble} -- a probabilistic procedure which iteratively constructs a matching in small bits -- can produce an almost-perfect matching in $H$, provided $D_2$ is much smaller than $D$. The quantitative aspects of this result were sharpened by several authors, with the previously best-known result due to Vu, whose result takes more of the codegree sequence $D_2, \dots, D_{k+1}$ into account. We improve Vu's result, by showing the Rödl Nibble can ``exhaust'' the full codegree sequence up to one of several natural bottlenecks, even tolerating extensive ``clustering'' of codegree values. Up to a subpolynomial error term, we believe our result to be the optimal usage of pure nibble methodology. We also show that our matching can be taken to be ``pseudorandom'' with respect to a set of weight functions on $V(H)$, and we use this result to derive other hypergraph matching results in partite settings, including a new bound on the list chromatic index which implies the best-known result of Molloy and Reed up to the error term, and is stronger when the hypergraph is not close to linear, i.e.\ $D_2=ω(1)$. We also apply our results to obtain improved bounds on almost-spanning structures in Latin squares and designs, and the maximum diameter of a simplicial complex.

Advancing the Rödl Nibble: New bounds on matchings and the list chromatic index of hypergraphs

TL;DR

This work advances the Rödl nibble framework for hypergraph matchings by exhaustively exploiting the full codegree sequence up to natural bottlenecks, enabling near-perfect coverings with subpolynomial errors. It introduces a robust pseudorandom matching paradigm via a family of weight functions, yielding applications to list colorings and to a spectrum of combinatorial structures including Latin squares, Steiner systems, and simplicial complexes. The core methodology combines a two-stage Nibble/Chomp process with concentration tools (including Linear Talagrand with exceptional outcomes and Kim–Vu polynomial concentration) and absorption-inspired reservoir techniques to convert almost-matchings into near-perfect or perfect ones in complex, multipartite settings. The results yield improved bounds on the list chromatic index, as well as near-optimal leftovers for diverse combinatorial objects, highlighting the practical impact of nibble-based approaches in design theory, Latin squares, and higher-dimensional combinatorial constructions.

Abstract

Let be a -uniform hypergraph which is nearly -regular, such that any set of vertices is contained in at most edges of for each . Influential results of Pippenger and of Frankl and Rödl show that the \textit{Rödl Nibble} -- a probabilistic procedure which iteratively constructs a matching in small bits -- can produce an almost-perfect matching in , provided is much smaller than . The quantitative aspects of this result were sharpened by several authors, with the previously best-known result due to Vu, whose result takes more of the codegree sequence into account. We improve Vu's result, by showing the Rödl Nibble can ``exhaust'' the full codegree sequence up to one of several natural bottlenecks, even tolerating extensive ``clustering'' of codegree values. Up to a subpolynomial error term, we believe our result to be the optimal usage of pure nibble methodology. We also show that our matching can be taken to be ``pseudorandom'' with respect to a set of weight functions on , and we use this result to derive other hypergraph matching results in partite settings, including a new bound on the list chromatic index which implies the best-known result of Molloy and Reed up to the error term, and is stronger when the hypergraph is not close to linear, i.e.\ . We also apply our results to obtain improved bounds on almost-spanning structures in Latin squares and designs, and the maximum diameter of a simplicial complex.

Paper Structure

This paper contains 29 sections, 34 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Rödl Nibble and previous results
  3. Our main result
  4. Pseudorandom hypergraph matchings
  5. Beyond matchings
  6. Applications
  7. Latin squares
  8. Simplicial complexes
  9. Designs
  10. Organisation of the paper
  11. Sketch of the proof of Theorem \ref{['theorem:maintheorem']}
  12. Preliminaries
  13. Notation
  14. Tools
  15. Beyond matchings
  16. ...and 14 more sections

Key Result

Theorem 1.1

Suppose that $1/D\ll1/K,1/k,\delta,\gamma<1$. Let $H$ be a $(k+1)$-uniform, $(n,D,{\varepsilon})$-regular hypergraph with $C_2(H)\leq D_2\leq D^{1-\gamma}$ and ${\varepsilon}\leq K\sqrt{\log D/D}$. Then there is a matching $M$ of $H$ which covers all but at most $n(D/D_2)^{-1/k +\delta}$ vertices.

Figures (1)

  • Figure 1: An illustration of semi-stuck, super-stuck, and slowpoke indices, and clustering of the degree sequence.

Theorems & Definitions (54)

  • Theorem 1.1: Kostochka and Rödl KR97, 1997
  • Theorem 1.2: Vu V00, 2000
  • Theorem 1.3: Kang, Kühn, Methuku, and Osthus KKMO23, 2023
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Bipartite Matching Theorem
  • Theorem 1.7
  • proof
  • Theorem 1.8: Kang, Kühn, Methuku, and Osthus KKMO23
  • Theorem 1.9: Molloy and Reed MR00
  • ...and 44 more