Tree-Matchings in Graph Processes

For a tree T a perfect T-matching in a graph G is a subgraph of G with at least $|G| - |T| + 1$ vertices, each component of which is isomorphic to T. Two properties, $\mathcal{A}$ and $\mathcal{B}$, are introduced where the former is a modification of the fact that the largest component of G has a perfect T-matching and the latter is a suitably chosen necessary condition for $\mathcal{A}$ expressed in terms of forbidden “pendant” subgraphs. We show that in the random graph process $\hat G_n $ the hitting times of both above properties coincide. This paper is the first one that deals with the hitting times of nonmonotone graph properties. It extends results of Bollobas and Frieze [Ann. Discrete Math., 28 (1985), pp. 23–46] and Bollobas and Thomason [Ann. Discrete Math., 28 (1985), pp. 47–98].