Nearly spanning cycle in the percolated hypercube

TL;DR

This paper proves that in a random subgraph of a hypercube, with sufficiently high edge retention probability, there almost certainly exists a cycle nearly spanning the entire hypercube, confirming a longstanding conjecture.

Contribution

It establishes that for any small positive epsilon, a random subgraph of the hypercube with edge probability above a certain threshold contains a nearly spanning cycle, resolving a major open problem.

Findings

High probability of near-spanning cycles in hypercube subgraphs

Threshold probability for cycle existence proportional to 1/d

Confirmation of a long-standing folklore conjecture

Abstract

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny D\'iaz, Gir\~ao, K\"uhn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534].