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Sufficient conditions for perfect mixed tilings

Eoin Hurley, Felix Joos, Richard Lang

TL;DR

A method to study sufficient conditions for perfect mixed tilings is developed and a conjecture of Komlós in a strong sense is resolved, which allows the embedding of bounded degree graphs H with components of sublinear order.

Abstract

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs $H$ with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect $F$-tilings (for an arbitrary fixed graph $F$) by replacing the $F$-tiling with the aforementioned graphs $H$. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.

Sufficient conditions for perfect mixed tilings

TL;DR

A method to study sufficient conditions for perfect mixed tilings is developed and a conjecture of Komlós in a strong sense is resolved, which allows the embedding of bounded degree graphs H with components of sublinear order.

Abstract

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect -tilings (for an arbitrary fixed graph ) by replacing the -tiling with the aforementioned graphs . Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.
Paper Structure (37 sections, 62 theorems, 40 equations)

This paper contains 37 sections, 62 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Guest graph properties
  3. Host graph properties
  4. Main result
  5. Organisation of the paper
  6. More details on the bigger picture
  7. Proof outline
  8. Notation
  9. Host graph properties
  10. Minimum Degree Conditions
  11. Degree sequences
  12. Universally dense graphs
  13. Guest graph properties
  14. Graphs in $\mathop{\mathrm{\mathcal{F}_{\rm {cr}}}}\nolimits$ are flexible
  15. Graphs of low critical chromatic number are flexible
  16. ...and 22 more sections

Key Result

Theorem 1.1

For all $\chi,\Delta > 1$, $\mu >0$ and $n$ sufficiently large, the following holds. Let $H$ be an $\mathop{\mathrm{\mathcal{F}_{\rm {cr}}}}\nolimits(\lceil \chi \rceil, \sqrt{n}/\log n)$-tiling on $n$ vertices with $\mathop{\mathrm{\chi_{\rm cr}}}\nolimits(H)\leqslant \chi$ and $\Delta(H) \leqslant

Theorems & Definitions (113)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5: Flexi-chromatic graphs
  • Definition 1.6: Tiling framework
  • Theorem 1.7: Tiling framework theorem
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 103 more