On the tree-depth of Random Graphs

TL;DR

This paper analyzes the asymptotic behavior of the tree-depth parameter in various models of random graphs, revealing phase transitions and confirming conjectures about its growth rate.

Contribution

It provides the first comprehensive asymptotic analysis of tree-depth in random graphs, confirming conjectures and establishing new bounds for different regimes.

Findings

Tree-depth is linear for dense graphs with p>>1/n.

Tree-depth is Theta(log n) at the critical point c=1.

Tree-depth is Theta(log log n) for sparse graphs with c<1.

Abstract

The tree-depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree-depth of random graphs. For dense graphs, p>> 1/n, the tree-depth of a random graph G is a.a.s. td(G)=n-O(sqrt(n/p)). Random graphs with p=c/n, have a.a.s. linear tree-depth when c>1, the tree-depth is Theta (log n) when c=1 and Theta (loglog n) for c<1. The result for c>1 is derived from the computation of tree-width and provides a more direct proof of a conjecture by Gao on the linearity of tree-width recently proved by Lee, Lee and Oum. We also show that, for c=1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree-depth.