Vertex covers by edge disjoint cliques

Let H be a simple graph having no isolated vertices. An (H, k)-vertex-cover of a simple graph G=(V, E) is a collection H1,...,Hr of subgraphs of G satisfying 1. Hi ≅ H, for all i = 1,...,r, 2. ∪ ri=1V(Hi)=V, 3. E(Hi) ∩ E(Hj) = 0, for all i≠j, and 4. each ν ∈ V is in at most k of the Hi. We consider the existence of such vertex covers when H is a complete graph, Kt,t≥3, in the context of extremal and random graphs.