Sparse pancyclic subgraphs of random graphs

It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high probability, $G(n,p)$ contains a pancyclic subgraph with $n+(1+o(1))\log_2 n$ edges for $p \ge p^*$, where $p^*=(1+o(1))\ln n/n$, right above the threshold for pancyclicity.