Random Enriched Trees with Applications to Random Graphs

Random Trees in Random Graphs

We show that a random labeled n-vertex graph almost surely contains isomorphic copies of almost all labeled n-vertex trees, in two senses. In the first sense, the probability of each edge occurring in the graph diminishes as n increases, and the set of trees referred to as "almost all" depends on the random graph. In the other sense, the probability of an edge occurring is fixed, and the releva...

Trees in random graphs

Let us consider the probability space Ifi(n, p) consisting of all graphs on n labeled vertices where each edge occurs with probability p =1q, independently of all other edges . The aim of this note is to find such natural numbers which are likely to occur as orders of maximal induced trees contained in a graph G E 19(n, p) when 0 < p < 1 is fixed . By a maximal induced tree we mean an induced t...

Random trees and applications

We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous’ theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applica...

Addendum to "trees in random graphs"

The aim of this addendum is to explain more precisely the second part of the proof of Theorem 1 from our paper [1] . We need to show that a.e. graph G E lfi(n, p) contains a maximal induced tree of order less than (1+6)x (log n)/(log d) . The second moment method used in our Lemma shows in fact that Prob{0 .9 E(X,) < X, < 1 .1 E(X,)} = 1-o(1) . (1') Now let S, stand for the number of (1, r)-sta...

Multicolored Trees in Random Graphs