On the tree-depth of random graphs

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 a random graph on n vertices where each edge appears independently with probability p. For dense graphs, np→+∞, the tree-depth of a random graph G is aastd(G)=n−O(n/p). Random graphs with p=c/n, have aaslinear tree-depth when c>1, the tree-depth is Θ(logn) when c=1 and Θ(loglogn) 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 (2012) [15]. We also show that, for c=1, every width parameter is aasconstant, and that random regular graphs have linear tree-depth.