Cycle lengths in the percolated hypercube

TL;DR

This paper proves that in a percolated hypercube with sufficiently high edge probability, the resulting subgraph almost surely contains cycles of all even lengths from 4 up to nearly the total number of vertices.

Contribution

The authors strengthen previous results by showing the existence of cycles of all even lengths within a certain range in the percolated hypercube.

Findings

Cycles of all even lengths between 4 and nearly 2^d exist with high probability

The result holds for percolation probabilities where pd exceeds a certain threshold

The work extends understanding of cycle structure in random subgraphs of hypercubes

Abstract

Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2^d$.