Hamilton cycles in 3-out

TL;DR

This paper proves that a random graph where each vertex chooses 3 neighbors uniformly at random almost surely contains a Hamilton cycle as the number of vertices grows large.

Contribution

It establishes that the 3-out random graph model is almost surely Hamiltonian for large n, advancing understanding of Hamiltonicity in sparse random graphs.

Findings

G_{3-out} is Hamiltonian with high probability as n→∞

Minimum degree 3 suffices for Hamiltonicity in this model

The probability of non-Hamiltonian 3-out graphs tends to zero

Abstract

Let G_{\rm 3-out} denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G_{\rm 3-out} has minimum degree 3 and average degree 6. We prove that the probability that G_{\rm 3-out} is Hamiltonian goes to 1 as n tends to infinity.