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Perfect tilings of 3-graphs with the generalised triangle

Candida Bowtell, Amarja Kathapurkar, Natasha Morrison, Richard Mycroft

TL;DR

This work resolves the optimal minimum codegree threshold for a perfect tiling of a $3$-uniform hypergraph by copies of the generalised triangle $T$, proving that for sufficiently large $n$ with $5 mid n$ there is no such tiling and for $5 mid n$ the threshold is tight at $ abla(H) \,=\, 2/5$. The authors develop a two-pronged proof, handling extremal configurations directly and applying an absorbing method combined with a tiling lemma that leverages fractional tilings via Farkas’ lemma to obtain almost-perfect tilings, which are then completed using a small absorbing set. They also extend the results to rainbow tilings, showing asymptotically optimal minimum codegree conditions for a perfect rainbow $T$-tiling and establishing, via Lang’s theorem, that the rainbow tiling threshold matches the non-rainbow one. The techniques introduce a novel case-analysis for fractional tilings and integrate closure/linkedness concepts with classical matching results (Daykin–Häggkvist) to obtain the exact threshold, advancing the understanding of tilings in non-tripartite $3$-graphs and providing a framework potentially adaptable to other fixed subgraphs.

Abstract

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$ and three edges $abc$, $abd$, $cde$. We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.

Perfect tilings of 3-graphs with the generalised triangle

TL;DR

This work resolves the optimal minimum codegree threshold for a perfect tiling of a -uniform hypergraph by copies of the generalised triangle , proving that for sufficiently large with there is no such tiling and for the threshold is tight at . The authors develop a two-pronged proof, handling extremal configurations directly and applying an absorbing method combined with a tiling lemma that leverages fractional tilings via Farkas’ lemma to obtain almost-perfect tilings, which are then completed using a small absorbing set. They also extend the results to rainbow tilings, showing asymptotically optimal minimum codegree conditions for a perfect rainbow -tiling and establishing, via Lang’s theorem, that the rainbow tiling threshold matches the non-rainbow one. The techniques introduce a novel case-analysis for fractional tilings and integrate closure/linkedness concepts with classical matching results (Daykin–Häggkvist) to obtain the exact threshold, advancing the understanding of tilings in non-tripartite -graphs and providing a framework potentially adaptable to other fixed subgraphs.

Abstract

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a -uniform hypergraph with copies of the generalised triangle , which is the 3-uniform hypergraph with five vertices and three edges , , . We also give an asymptotically-optimal minimum codegree condition for the rainbow version of the problem.
Paper Structure (10 sections, 19 theorems, 10 equations, 1 figure)

This paper contains 10 sections, 19 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Perfect tilings in graphs and hypergraphs
  3. The generalised triangle
  4. Proof strategy
  5. Rainbow tilings
  6. Notation
  7. The extremal case
  8. Absorbing
  9. Almost-perfect tilings
  10. Perfect rainbow tilings

Key Result

Theorem 1.1

There exists $n_0 \in \mathbb{N}$ such that for every $n \geq n_0$ with $5 \mid n$, every $3$-uniform hypergraph $H$ on $n$ vertices with $\delta(H) \geq 2n/5$ admits a perfect $T$-tiling.

Figures (1)

  • Figure 1: The generalised triangle $3$-graph $T$.

Theorems & Definitions (31)

  • Theorem 1.1: Main result
  • Definition 1.2
  • Lemma 1.3: Extremal case
  • Lemma 1.4: Absorbing lemma
  • Lemma 1.5: Tiling lemma
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem 1.6
  • Theorem 2.1: Daykin and Häggkvist DH
  • Corollary 2.2
  • proof
  • ...and 21 more