On the order of the largest induced tree in a random graph

AbstractConsider a random graph K(n, p) with n labeled vertices in which the edges are chosen independently and with a probability p. Let Tn(p) be the order of the largest induced tree in K(n, p). Among other results it is shown, using an algorithmic approach, that if p=(c log n)/n, where c ≥ e is a constant, then for any fixed ε > 01c−εlog lognlognn<Tn(p)<2c+εlog lognlogn almost surely