Cycle lengths in the percolated hypercube

Abstract

Let $Q_p^d$ 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 $\epsilon >0$, there is some $c=c(\epsilon )>0$ such that, if $pd\ge c$, then typically $Q_p^d$ contains a cycle of length at least $(1-\epsilon )2^d$. We strengthen this result to show that, under the same assumptions, typically $Q_p^d$ contains cycles of all even lengths between $4$ and $(1-\epsilon )2^d$.