Let t > 1 be an integer. A (t,2)-vertex-cover of a graph G = (V,E) is a collection of edge disjoint subgraphs of G, each isomorphic to the complete graph K_t, such that the collection covers every vertex in V either once or twice. Consider the graph process G₀, G₁, G₂,... in which we start with the empty graph and add edges (chosen uniformly at random) one at a time. We consider two hitting times. Let T₁ be the least m for which every vertex in G_m is contained in a copy of K_t, and let T₂ be the least m for which G_m has a (t,2)-vertex-cover. We show that with high probability T₁ = T₂. This result is related to a well known conjecture of Erdos and Spencer concerning the existence of K₃-factors in the random graph. (Joint work with T. Bohman, A. Frieze, and M. Ruszinko.)