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Cycle decompositions in $k$-uniform hypergraphs

Allan Lo, Simón Piga, Nicolás Sanhueza-Matamala

Abstract

We show that $k$-uniform hypergraphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths. In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary $k$-uniform hypergraphs, which should be of independent interest.

Cycle decompositions in $k$-uniform hypergraphs

Abstract

We show that -uniform hypergraphs on vertices whose codegree is at least can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths. In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary -uniform hypergraphs, which should be of independent interest.
Paper Structure (27 sections, 40 theorems, 110 equations, 1 figure)

This paper contains 27 sections, 40 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof ideas
  3. Organisation
  4. Notation
  5. Lower bounds
  6. Absorbers versus transformers
  7. Iterative absorption and proof of the main result
  8. Vortex lemma
  9. Transformers I: gadgets
  10. Residual graphs
  11. Basic gadgets
  12. Balancer gadgets
  13. Swapper gadgets
  14. Transformers II: Tour-trail decompositions
  15. Basic properties
  16. ...and 12 more sections

Key Result

Theorem 1.3

For every $k\geq 3$ there exists an $\ell_0\in\mathbb{N}$ such that for every $\ell\geq \ell_0$ it holds that $\delta_{{C_\ell^{(k)}}}\leq 2/3$.

Figures (1)

  • Figure 1: The tour-trail decomposition $\mathcal{T}_j$ of the basic gadget $G = G_j(\mathbf y, x,x')$ and its residual di-$(k-1)$-graph $D(\mathcal{T}_j)$. Dotted lines represent tight paths using new vertices.

Theorems & Definitions (80)

  • Conjecture 1.1: Glock, Kühn and Osthus
  • Conjecture 1.2: Glock, Kühn, and Osthus GlockKuhnOsthus2021
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • proof : Proof of Proposition \ref{['proposition:newlowerbound']}
  • Corollary 2.3
  • ...and 70 more