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A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

Amarja Kathapurkar, Patrick Morris, Guillem Perarnau

TL;DR

This work settles a hypergraph rainbow-Dirac-type question for loose Hamilton cycles: above the j-degree threshold δ_j(G) ≥ (δ_j^k(1) + ε) n^{k−j}, every globally bounded colouring of a large k-graph G contains a rainbow loose Hamilton cycle. The authors deploy absorption and a sophisticated switching framework based on random m-splittings to convert local alterations into a global rainbow structure, controlled via the lopsided Lovász Local Lemma and concentration inequalities. A trio of core lemmas connects (i) the abundance of feasible switchings, (ii) the existence of suitable splittings with viable partitions, and (iii) the construction of many such splittings via random sampling, with an absorption-step ensuring coherence of the final rainbow cycle. The results support the meta-conjecture that Dirac-type extremal thresholds exhibit rainbow robustness under bounded colourings in hypergraphs and point toward extensions to tight cycles and linear F-factors, albeit with notable obstacles in non-loose settings.

Abstract

A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

TL;DR

This work settles a hypergraph rainbow-Dirac-type question for loose Hamilton cycles: above the j-degree threshold δ_j(G) ≥ (δ_j^k(1) + ε) n^{k−j}, every globally bounded colouring of a large k-graph G contains a rainbow loose Hamilton cycle. The authors deploy absorption and a sophisticated switching framework based on random m-splittings to convert local alterations into a global rainbow structure, controlled via the lopsided Lovász Local Lemma and concentration inequalities. A trio of core lemmas connects (i) the abundance of feasible switchings, (ii) the existence of suitable splittings with viable partitions, and (iii) the construction of many such splittings via random sampling, with an absorption-step ensuring coherence of the final rainbow cycle. The results support the meta-conjecture that Dirac-type extremal thresholds exhibit rainbow robustness under bounded colourings in hypergraphs and point toward extensions to tight cycles and linear F-factors, albeit with notable obstacles in non-loose settings.

Abstract

A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any , if is a -uniform hypergraph above the -degree threshold for a loose Hamilton cycle, then any globally bounded colouring of contains a rainbow loose Hamilton cycle.
Paper Structure (13 sections, 15 theorems, 41 equations, 4 figures)

This paper contains 13 sections, 15 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rainbow spanning structures
  3. Rainbow structures in Dirac (hyper-)graphs
  4. Organisation
  5. Preliminaries
  6. Notation and terminology
  7. Probabilistic tools
  8. Proof Overview
  9. Many feasible switchings imply a rainbow Hamilton cycle
  10. Suitable splittings with viable partitions give feasible switchings
  11. Path tilings
  12. Many suitable splittings with viable partitions
  13. Concluding Remarks

Key Result

Theorem 1.2

Let $1\leq j< k\in \mathbb N$ and let $\varepsilon>0$ be arbitrary. Then there exists $\mu>0$ such that for any sufficiently large $n\in (k-1)\mathbb N$, we have that if $G$ is an $n$-vertex $k$-graph with $\delta_j(G)\geq (\delta_j^k(1)+\varepsilon)n^{k-j}$, the following holds. For any colouring $ we have that there is a rainbow loose Hamilton cycle in $G$ coloured by $\chi$.

Figures (4)

  • Figure 1: On the left, an example of a transverse partition ${\mathcal{Y}}=\{Y_1,Y_2,Y_3,Y_4\}$ with paths $P_0,\ldots,P_7$ (which are just single edges of a $4$-uniform hypergraph in this case) such that $V(P_i)=\{u^i_a, u^i_b, u^i_c,u^i_d\}$ (denoted in the figure simply as $a,b,c,d$ in each row) and $u^i_a$ and $u^i_b$ are the endpoints of $P_i$. On the right, the digraph $\stackon[-4.2pt]{K_8}{\,{\stackon[-1.95pt]{\mathchar"017E}{\hbox{$\mathchar"017E$}}}}\setminus \vec{D}$ where $\stackon[-4.2pt]{K_8}{\,{\stackon[-1.95pt]{\mathchar"017E}{\hbox{$\mathchar"017E$}}}}$ is the complete digraph on $8$ vertices with all edges $\vec{ii'}$ with $0\leq i<i'\leq 8$, and $\vec{D}$ is the auxiliary digraph defined by ${\mathcal{Y}}$.
  • Figure 2: An example of Hamilton dicycle $\vec{H}$ in the digraph $\vec{D}$ given by the partition ${\mathcal{Y}}$ given in Figure \ref{['fig:partition1']}.
  • Figure 3: On the left, the pairs ${\mathcal{R}}$ are identified for the partition ${\mathcal{Y}}$ according to the Hamilton dicycle $\vec{H}$ from Figure \ref{['fig:Hamdi']}. The values of $h_i$ and the vertices $w_i$ are then given and the resulting partition ${\mathcal{X}}$.
  • Figure 4: The construction for Example \ref{['example:tight']} in the case that $n\in 3 \mathbb N$.

Theorems & Definitions (39)

  • Definition 1.1: Hypergraph Hamilton cycle thresholds
  • Theorem 1.2: Main Theorem
  • Lemma 2.1: The lopsided local lemma
  • Remark 2.2
  • Definition 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5: Chernoff inequality
  • Lemma 2.6
  • Lemma 2.7: Suen's inequality
  • ...and 29 more