ON LARGE INDUCED TREES AND LONG INDUCED PATHS IN SPARSE RANDOM GRAPHS

Let Gn,p denote the graph obtained from deleting the edges of Kn, the complete graph with vertex set Vn = {1,2, ..., n}, independently with equal probability 1 - p. Assume that p = p(n) is such that np = c > 1. We describe an algorithm FindTree for finding induced trees in a graph. By analyzing how FindTree performs in Gn,p, we obtain the following results. Let Tn be the order of the largest induced subtree of Gn,p. We find a number t(c) such that Tn is almost surely larger than (t(c) - ε)n for any ε > 0. Also, if Ln denotes the length of the longest induced path in Gn,p, then we find a number h(c) such that Ln is almost surely larger than (h(c) - ε)n for any ε > 0. © 1992.