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Colorful Hamilton cycles in random graphs

Debsoumya Chakraborti, Alan Frieze, Mihir Hasabnis

Abstract

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges of color $c_i$. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r: m_1+m_2+\cdots+m_r=n\}$.

Colorful Hamilton cycles in random graphs

Abstract

Given an vertex graph whose edges have colored from one of colors , we define the Hamilton cycle color profile to be the set of vectors such that there exists a Hamilton cycle that is the concatenation of paths , where contains edges of color . We study when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when .

Paper Structure

This paper contains 20 sections, 14 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1']}
  4. Validation of Step 1
  5. Validation of Step 2
  6. Expansion properties
  7. Step 2: Constructing Hamilton paths in $G_i$
  8. Step 3: Connecting the Hamilton paths together
  9. Proof of Theorem \ref{['th2']}
  10. Step 1: Construction of disjoint $Q_v$'s
  11. Step 2: Construction of paths $P_1,P_2,\ldots,P_{r-1}$
  12. Step 3: Construction of $P_r$ and the Hamilton cycle
  13. Proof of Theorems \ref{['th3']} and \ref{['th4']}
  14. Structural lemmas
  15. Proof of Lemma \ref{['lem0']}
  16. ...and 5 more sections

Key Result

Theorem 1

Fix $r \ge 2$ and positive real numbers $\beta,\alpha_1,\alpha_2,\ldots,\alpha_r$ where $\sum_{i=1}^r \alpha_i = 1$. If $p \ge \frac{\log n+r\log\log n+\omega}{n}$ where $\omega = \omega(n) \rightarrow \infty$ as $n \rightarrow \infty$, then w.h.p. $hcp(G_{n,p}^{\boldsymbol \alpha})\supseteq {\bf M}

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 8 more