Limiting shape of the depth first search tree in an Erdos-Renyi graph

We show that the profile of the tree constructed by the depth first search algorithm in the giant component of an Erdos-Renyi graph with N vertices and connection probability c/N with c > 1 converges to an explicit deterministic shape. This makes it possible to exhibit a long nonintersecting path of length rho c-Li2(rho c)cxN, where rho(c) is the density of the giant component and Li-2 denotes the dilogarithm function.