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Limiting shape of the depth first search tree in an Erdős‐Rényi graph

We show that the profile of the tree constructed by the depth first search algorithm in the giant component of an Erdős‐Rényi 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 ρc−Li2(ρc)c×N , where ρc is the density of the giant component and Li2 denotes the dilogarithm function.