Hamiltonicity of random subgraphs of the hypercube

TL;DR

This paper proves that random subgraphs of the hypercube almost surely contain multiple edge-disjoint Hamilton cycles once a certain degree threshold is reached, resolving a longstanding conjecture about the hypercube's Hamiltonicity threshold.

Contribution

It establishes optimal hitting time and perturbation results for Hamiltonicity in hypercube subgraphs, confirming the conjecture that the threshold probability is 1/2.

Findings

Random subgraphs with minimum degree 2k contain k edge-disjoint Hamilton cycles.

The threshold probability for Hamiltonicity in the hypercube is 1/2.

Random subgraphs contain almost spanning cycles with high probability.

Abstract

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph $\mathcal{Q}^n_p$ of $\mathcal{Q}^n$ with $p\in(0,1]$ fixed, then with high probability $H\cup\mathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. In particular, both results resolve a long standing conjecture, posed e.g. by Bollob\'as, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. Our techniques also show that, with high probability, for all fixed $p\in(0,1]$ the graph $\mathcal{Q}^n_p$ contains an almost spanning cycle. Our methods involve branching processes, the R\"odl nibble, and absorption.