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Rainbow powers of a Hamilton cycle in G(n,p)

Tolson Bell, Alan Frieze

Abstract

We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of $G_{n,p}$ is within a constant factor of the uncolored threshold. Our proof requires $(1+\varepsilon)$ times the minimum number of colors.

Rainbow powers of a Hamilton cycle in G(n,p)

Abstract

We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of is within a constant factor of the uncolored threshold. Our proof requires times the minimum number of colors.
Paper Structure (8 sections, 6 theorems, 30 equations)

This paper contains 8 sections, 6 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Some notation
  3. Proof of Theorem \ref{['power']}
  4. The size of ${\mathcal{H}}^*$
  5. Random sample from $X$
  6. Finishing the proof
  7. Powers of Hamilton cycles
  8. Final thoughts

Key Result

Theorem 1

Let ${\mathcal{H}}$ be an $r$-bounded, $\kappa$-spread hypergraph and let $X=V({\mathcal{H}})$. There is an absolute constant $K>0$ such that if then w.h.p. $X_p$ or $X_m$ respectively contains an edge of ${\mathcal{H}}$. More precisely, $\mathbb{P}(X_p\text{ contains an edge of ${\mathcal{H}}$ })\geq 1-\varepsilon_r$ where $\varepsilon_r\to0$ as $r\to\infty$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof