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Exact minimum co-degree conditions for $\ell$-Hamiltonicity in hypergraphs

Luyining Gan, Jie Han, Huan Xu

Abstract

Suppose $1\le \ell <k$ such that $(k-\ell)\nmid k$. Given an $n$-vertex $k$-uniform hypergraph $\mathcal H$, for all $k/2<\ell< 3k/4$ and sufficiently large $n\in (k-\ell)\mathbb N$, we prove that if $\mathcal H$ has minimum co-degree at least $\frac{n}{\lceil \frac{k}{k-\ell}\rceil (k-\ell)}$, then $\mathcal H$ contains a Hamilton $\ell$-cycle, which partially verifies a conjecture of Han and Zhao and (partially) resolves a problem of Rödl and Ruciński. Moreover, we show that assuming minimum co-degree $\frac{n}{\lceil \frac{k}{k-\ell}\rceil (k-\ell)}+\frac{k^2}2$ is enough for all $\ell$.

Exact minimum co-degree conditions for $\ell$-Hamiltonicity in hypergraphs

Abstract

Suppose such that . Given an -vertex -uniform hypergraph , for all and sufficiently large , we prove that if has minimum co-degree at least , then contains a Hamilton -cycle, which partially verifies a conjecture of Han and Zhao and (partially) resolves a problem of Rödl and Ruciński. Moreover, we show that assuming minimum co-degree is enough for all .
Paper Structure (14 sections, 21 theorems, 51 equations, 2 figures)

This paper contains 14 sections, 21 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Hightlights of proof ideas
  4. Auxiliary lemmas and proof of Theorem \ref{['thm:nonextremal']}
  5. Auxiliary lemmas
  6. Proof of Theorem \ref{['thm:nonextremal']}
  7. Proof of Theorem \ref{['thm:extremal']}
  8. Preparation
  9. Useful tools
  10. Proof of Theorem \ref{['thm:main result']}
  11. Building a short $\ell$-path ${\mathcal{Q}}$
  12. Completing the Hamilton $\ell$-cycle
  13. Finding $sq$ vertex-disjoint $\ell$-paths under minimum co-degree conditions
  14. Remarks

Key Result

Theorem 1.3

(Exact result for $\ell<3k/4$). Let integers $k\geq 3$ and $k/2< \ell<3k/4$ such that $(k-\ell)\nmid k$. Assume that $n\in (k-\ell)\mathbb{N}$ is sufficiently large. If ${\mathcal{H}}=(V,E)$ is an $n$-vertex $k$-graph such that then ${\mathcal{H}}$ contains a Hamilton $\ell$-cycle.

Figures (2)

  • Figure 1: Extensions from an $\ell$-end $L_0$ and the relevant sets for $k=7, \ell=5$
  • Figure 2: $\ell$-path contains $v$ right in the middle of the path for $k=7$, $\ell=5$

Theorems & Definitions (43)

  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Theorem 2.2: Weak Regularity Lemma
  • Proposition 2.3
  • Lemma 2.4
  • ...and 33 more