# Limiting shape of the Depth First Search tree in an Erd\H{o}s-R\'enyi graph

## TL;DR

This paper characterizes the limiting shape of the DFS tree in Erdős-Rényi graphs, revealing a deterministic profile and identifying a long non-intersecting path related to the giant component's density.

## Contribution

It provides an explicit deterministic shape for the DFS tree in Erdős-Rényi graphs and demonstrates the existence of a long non-intersecting path proportional to the graph size.

## Key findings

- DFS tree profile converges to a deterministic shape
- Existence of a long non-intersecting path of specified length
- Explicit relation involving the giant component density

## Abstract

We show that the profile of the tree constructed by the Depth First Search Algorithm in the giant component of an Erd\H{o}s-R\'enyi graph with $N$ vertices and connection probability $c/N$ converges to an explicit deterministic shape. This makes it possible to exhibit a long non-intersecting path of length $\left( \rho_c - \frac{\mathrm{Li}_2(\rho_c)}{c} \right) \times N$, where $\rho_c$ is the density of the giant component.