Abstract. It is known that the complete graph [Formula: see text] contains a pancyclic subgraph with [Formula: see text] edges, and that there is no pancyclic graph on [Formula: see text] vertices with fewer than [Formula: see text] edges. We show that, with high probability, [Formula: see text] contains a pancyclic subgraph with [Formula: see text] edges for [Formula: see text], where [Formula: see text], which is right above the threshold for pancyclicity.

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. there is a positive constant $\varepsilon > 0$ such that $\delta (G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. there are some constants $\alpha, \beta >0$ such that every two disjoint subsets $U,W$ of size at least $\alpha n$ have at least $\beta |U||W|$ edges between them, and the minimum degree of $G$ is at least $(2\alpha + \beta )\cdot n$; or 3. $G$ is an $(n,d,\lambda )$-graph, with $d\geq \frac{C\cdot n\cdot \log \log n}{\log n}$ and $\lambda \leq \frac{c\cdot d^2}{n}$ for some absolute constants $c,C>0;$ then for a positive integer constant $k$ with high probability the hitting time of the property of containing $k$ edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least $2k$. These results extend prior results by Johansson and by Frieze and Krivelevich and answer a question posed by Frieze.

We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.