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Positive codegree thresholds for Hamilton cycles in hypergraphs

Richard Mycroft, Camila Zárate-Guerén

TL;DR

The paper determines an asymptotically optimal minimum positive codegree threshold for guaranteeing a Hamilton $\ell$-cycle in a $k$-uniform hypergraph, for all $k\ge 3$ and $1\le \ell\le k-1$. It employs the absorbing method, a path-tiling lemma via the weak hypergraph regularity framework, and a weighted perfect fractional matching on a cluster hypergraph to achieve a near-spanning tiling and final absorption into a Hamilton cycle. The threshold $\delta^+(H) \ge \left(1-\frac{1}{\left\lfloor{\frac{k}{k-\ell}}\right\rfloor(k-\ell)}\right)n + \alpha n$ is shown to be asymptotically best possible (up to $\alpha n$) and generalizes known results, including the tight-cycle case $\ell = k-1$ related to ILMPS. The work also discusses a duality with minimum codegree thresholds and outlines future directions toward exact thresholds and extensions to $F$-tilings, aided by a novel weighted fractional matching approach. This advances the understanding of spanning structures in hypergraphs under positive codegree constraints and provides a versatile toolkit for related problems.

Abstract

For each $k \geq 3$ and $1 \leq \ell \leq k-1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $\ell$-cycle in a $k$-uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case $\ell = k-1$ of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, Müyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.

Positive codegree thresholds for Hamilton cycles in hypergraphs

TL;DR

The paper determines an asymptotically optimal minimum positive codegree threshold for guaranteeing a Hamilton -cycle in a -uniform hypergraph, for all and . It employs the absorbing method, a path-tiling lemma via the weak hypergraph regularity framework, and a weighted perfect fractional matching on a cluster hypergraph to achieve a near-spanning tiling and final absorption into a Hamilton cycle. The threshold is shown to be asymptotically best possible (up to ) and generalizes known results, including the tight-cycle case related to ILMPS. The work also discusses a duality with minimum codegree thresholds and outlines future directions toward exact thresholds and extensions to -tilings, aided by a novel weighted fractional matching approach. This advances the understanding of spanning structures in hypergraphs under positive codegree constraints and provides a versatile toolkit for related problems.

Abstract

For each and we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton -cycle in a -uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, Müyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.
Paper Structure (9 sections, 17 theorems, 17 equations)

This paper contains 9 sections, 17 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Connection to previous work
  3. Minimum codegree conditions for Hamilton cycles
  4. Minimum positive codegree conditions
  5. Discussion
  6. Notation and preliminaries
  7. Optimality of Theorem \ref{['main']}
  8. Proof of Theorem \ref{['main']}
  9. Path tiling

Key Result

Theorem 1.1

For all $k \geq 3$, $1\leq \ell \leq k- 1$ and $\alpha > 0$ there exists $n_0$ such that the following holds for every $n \geq n_0$ which is divisible by $k-\ell$. If $H$ is a $k$-graph on $n$ vertices with and $H$ has no isolated vertices, then $H$ contains a Hamilton $\ell$-cycle.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2: hh-han10hh-keevash11KMO14pp-kuhn06-cherryhh-rodl06hh-rodl08
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • proof
  • Proposition 1.8
  • ...and 20 more