Maximal induced trees in sparse random graphs

Abstract

A study of the orders of maximal induced trees in a random graph Gp with small edge probability p is given. In particular, it is shown that the giant component of almost every Gp, where p=c/n and c > 1 is a constant, contains only very small maximal trees (that are of a specific type) and very large maximal trees. The presented results provide an elementary proof of a conjecture from [3] that was confirmed recently in [4] and [5].