The threshold probability for long cycles

Abstract

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p\ge \frac{\log k+\log \log k+\ddot{I}{‰}_k(1)}{k}$, where $\ddot{I}{‰}_k(1)$ is any function tending to infinity with $k$, then $G_p$ asymptotically almost surely contains a cycle of length at least $k+1$. When we take $G$ to be the complete graph on $k+1$ vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.