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Hamiltonicity of inhomogeneous random graphs

Frederik Garbe, Jan Hladký, Simón Piga

Abstract

We provide a complete characterization of those graphons $W$ for which the inhomogeneous random graph $G(n,W)$ is asymptotically almost surely Hamiltonian. The characterization involves three conditions. Two of them constitute the characterization of $G(n,W)$ being a.a.s. connected, as was shown recently by Hladký and Viswanathan. The third condition captures a geometric obstacle which prevents $G(n,W)$ from having perfect fractional matchings.

Hamiltonicity of inhomogeneous random graphs

Abstract

We provide a complete characterization of those graphons for which the inhomogeneous random graph is asymptotically almost surely Hamiltonian. The characterization involves three conditions. Two of them constitute the characterization of being a.a.s. connected, as was shown recently by Hladký and Viswanathan. The third condition captures a geometric obstacle which prevents from having perfect fractional matchings.

Paper Structure

This paper contains 24 sections, 14 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Peninsulae
  3. Main result
  4. Necessity of condition \ref{['en:mainHCnegative1']}
  5. Necessity of condition \ref{['en:mainHCnegative2']}
  6. Necessity of condition \ref{['en:mainHCnegative3']}
  7. Outline of the proof
  8. Preliminaries
  9. Graphs
  10. Fractional matchings and fractional vertex covers in graphs
  11. Graphons
  12. Probability theory
  13. Functional analysis
  14. Asymptotic notation
  15. Regularity lemma
  16. ...and 9 more sections

Key Result

Theorem 1.2

Suppose that $W:\Omega^2\to[0,1]$ is a graphon. Then $\mathbb{G}(n,W)$ is a.a.s. Hamiltonian if the following three conditions are fulfilled: Further, these conditions are necessary: $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Left: A narrow graph peninsula; dark regions indicate admissible edges. Right: A beautiful peninsula. Included for pleasure.
  • Figure 2: Graphons $U$ and $W$ from Section \ref{['sssec:peninsula']}

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • Proposition 2.8: Theorem 1.10(i) in ConnectednessGraphons
  • ...and 16 more