in this talk we show how analyzing basic depth first search algorithm leads to a simple proof of the two most classical results about random graphs: 1. the result of erdos-renyi that the random graph $g(n,p)$ experiences sharp phase transition around $p=1/n$. for $p=(1-\epsilon)/n$, all connected components of $g(n,p)$ are typically of size $o(\log n)$, while for $p=(1+\epsilon)/n$, with high probability there exists a connected component of size linear in n. 2. the result of ajtai-komlos-szemeredi that in the supercritical regime $p=(1+\epsilon)/n$, random graph typically contains a path of linear length. joint work with m. krivelelvich